Factoring - Anna Kuczynska...section F1 | 217 Greatest Common Factor and Factoring by Grouping d. T...
Transcript of Factoring - Anna Kuczynska...section F1 | 217 Greatest Common Factor and Factoring by Grouping d. T...
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section F1 | 214
Factoring
Factoring
Factoring is the reverse process of multiplication. Factoring polynomials in algebra has similar role as factoring numbers in arithmetic. Any number can be expressed as a product of prime numbers. For example, 6 = 2 โ 3. Similarly, any polynomial can be expressed as a product of prime polynomials, which are polynomials that cannot be factored any further. For example, ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 6 = (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ + 3). Just as factoring numbers helps in simplifying or adding fractions, factoring polynomials is very useful in
simplifying or adding algebraic fractions. In addition, it helps identify zeros of polynomials, which in turn allows for solving higher degree polynomial equations.
In this chapter, we will examine the most commonly used factoring strategies with particular attention to special factoring. Then, we will apply these strategies in solving polynomial equations.
F.1 Greatest Common Factor and Factoring by Grouping
Prime Factors
When working with integers, we are often interested in their factors, particularly prime factors. Likewise, we might be interested in factors of polynomials.
Definition 1.1 To factor a polynomial means to write the polynomial as a product of โsimplerโ polynomials. For example,
5๐ฅ๐ฅ + 10 = 5(๐ฅ๐ฅ + 2), or ๐ฅ๐ฅ2 โ 9 = (๐ฅ๐ฅ + 3)(๐ฅ๐ฅ โ 3).
In the above definition, โsimplerโ means polynomials of lower degrees or polynomials with coefficients that do not contain common factors other than 1 or โ1. If possible, we would like to see the polynomial factors, other than monomials, having integral coefficients and a positive leading term.
When is a polynomial factorization complete?
In the case of natural numbers, the complete factorization means a factorization into prime numbers, which are numbers divisible only by their own selves and 1. We would expect that similar situation is possible for polynomials. So, which polynomials should we consider as prime?
Observe that a polynomial such as โ4๐ฅ๐ฅ + 12 can be written as a product in many different ways, for instance
โ(4๐ฅ๐ฅ + 12), 2(โ2๐ฅ๐ฅ + 6), 4(โ๐ฅ๐ฅ + 3), โ4(๐ฅ๐ฅ โ 3), โ12 ๏ฟฝ13๐ฅ๐ฅ + 1๏ฟฝ, etc.
Since the terms of 4๐ฅ๐ฅ + 12 and โ2๐ฅ๐ฅ + 6 still contain common factors different than 1 or โ1, these polynomials are not considered to be factored completely, which means that they should not be called prime. The next two factorizations, 4(โ๐ฅ๐ฅ + 3) and โ4(๐ฅ๐ฅ โ 3) are both complete, so both polynomials โ๐ฅ๐ฅ + 3 and ๐ฅ๐ฅ โ 3 should be considered as prime. But what about the last factorization, โ12 ๏ฟฝ1
3๐ฅ๐ฅ + 1๏ฟฝ? Since the remaining binomial 1
3๐ฅ๐ฅ + 1
does not have integral coefficients, such a factorization is not always desirable.
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Greatest Common Factor and Factoring by Grouping
Here are some examples of prime polynomials:
any monomials such as โ2๐ฅ๐ฅ2, ๐๐๐๐2, or 13๐ฅ๐ฅ๐ฅ๐ฅ;
any linear polynomials with integral coefficients that have no common factors other than 1 or โ1, such as ๐ฅ๐ฅ โ 1 or 2๐ฅ๐ฅ + 5;
some quadratic polynomials with integral coefficients that cannot be factored into any lower degree polynomials with integral coefficients, such as ๐ฅ๐ฅ2 + 1 or ๐ฅ๐ฅ2 + ๐ฅ๐ฅ + 1.
For the purposes of this course, we will assume the following definition of a prime polynomial.
Definition 1.2 A polynomial with integral coefficients is called prime if one of the following conditions is true
- it is a monomial, or - the only common factors of its terms are ๐๐ or โ๐๐ and it cannot be factored into any
lower degree polynomials with integral coefficients.
Definition 1.3 A factorization of a polynomial with integral coefficients is complete if all of its factors are prime.
Here is an example of a polynomial factored completely:
โ6๐ฅ๐ฅ3 โ 10๐ฅ๐ฅ2 + 4๐ฅ๐ฅ = โ2๐ฅ๐ฅ(3๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 2)
In the next few sections, we will study several factoring strategies that will be helpful in finding complete factorizations of various polynomials.
Greatest Common Factor
The first strategy of factoring is to factor out the greatest common factor (GCF).
Definition 1.4 The greatest common factor (GCF) of two or more terms is the largest expression that is a factor of all these terms.
In the above definition, the โlargest expressionโ refers to the expression with the most factors, disregarding their signs.
To find the greatest common factor, we take the product of the least powers of each type of common factor out of all the terms. For example, suppose we wish to find the GCF of the terms
6๐ฅ๐ฅ2๐ฅ๐ฅ3, โ18๐ฅ๐ฅ5๐ฅ๐ฅ, and 24๐ฅ๐ฅ4๐ฅ๐ฅ2.
First, we look for the GCF of 6, 18, and 24, which is 6. Then, we take the lowest power out of ๐ฅ๐ฅ2, ๐ฅ๐ฅ5, and ๐ฅ๐ฅ4, which is ๐ฅ๐ฅ2. Finally, we take the lowest power out of ๐ฅ๐ฅ3, ๐ฅ๐ฅ, and ๐ฅ๐ฅ2, which is ๐ฅ๐ฅ. Therefore,
GCF(6๐ฅ๐ฅ2๐ฅ๐ฅ3 , โ 18๐ฅ๐ฅ5๐ฅ๐ฅ, 24๐ฅ๐ฅ4๐ฅ๐ฅ2) = 6๐ฅ๐ฅ2๐ฅ๐ฅ
This GCF can be used to factor the polynomial 6๐ฅ๐ฅ2๐ฅ๐ฅ3 โ 18๐ฅ๐ฅ5๐ฅ๐ฅ + 24๐ฅ๐ฅ4๐ฅ๐ฅ2 by first seeing it as
6๐ฅ๐ฅ2๐ฅ๐ฅ โ ๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ2๐ฅ๐ฅ โ 3๐ฅ๐ฅ3 + 6๐ฅ๐ฅ2๐ฅ๐ฅ โ 4๐ฅ๐ฅ2๐ฅ๐ฅ,
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Factoring
and then, using the reverse distributing property, โpullingโ the 6๐ฅ๐ฅ2๐ฅ๐ฅ out of the bracket to obtain
6๐ฅ๐ฅ2๐ฅ๐ฅ(๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ3 + 4๐ฅ๐ฅ2๐ฅ๐ฅ).
Note 1: Notice that since 1 and โ1 are factors of any expression, the GCF is defined up to the sign. Usually, we choose the positive GCF, but sometimes it may be convenient to choose the negative GCF. For example, we can claim that
GCF(โ2๐ฅ๐ฅ,โ4๐ฅ๐ฅ) = 2 or GCF(โ2๐ฅ๐ฅ,โ4๐ฅ๐ฅ) = โ2,
depending on what expression we wish to leave after factoring the GCF out:
โ2๐ฅ๐ฅ โ 4๐ฅ๐ฅ = 2โ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐บ๐บ๐บ๐บ๐บ๐บ
(โ๐ฅ๐ฅ โ 2๐ฅ๐ฅ)๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ก๐ก๐ก๐ก
or โ2๐ฅ๐ฅ โ 4๐ฅ๐ฅ = โ2๏ฟฝ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐บ๐บ๐บ๐บ๐บ๐บ
(๐ฅ๐ฅ + 2๐ฅ๐ฅ)๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐๐ก๐ก๐ก๐ก
Note 2: If the GCF of the terms of a polynomial is equal to 1, we often say that these terms do not have any common factors. What we actually mean is that the terms do not have a common factor other than 1, as factoring 1 out does not help in breaking the original polynomial into a product of simpler polynomials. See Definition 1.1.
Finding the Greatest Common Factor
Find the Greatest Common Factor for the given expressions.
a. 6๐ฅ๐ฅ4(๐ฅ๐ฅ + 1)3, 3๐ฅ๐ฅ3(๐ฅ๐ฅ + 1), 9๐ฅ๐ฅ(๐ฅ๐ฅ + 1)2 b. 4๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ), 8๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ) c. ๐๐๐๐2, ๐๐2๐๐, ๐๐, ๐๐ d. 3๐ฅ๐ฅโ1๐ฅ๐ฅโ3, ๐ฅ๐ฅโ2๐ฅ๐ฅโ2๐ง๐ง
a. Since GCF(6, 3, 9) = 3, the lowest power out of ๐ฅ๐ฅ4, ๐ฅ๐ฅ3, and ๐ฅ๐ฅ is ๐ฅ๐ฅ, and the lowest power out of (๐ฅ๐ฅ + 1)3, (๐ฅ๐ฅ + 1), and (๐ฅ๐ฅ + 1)2 is (๐ฅ๐ฅ + 1), then
GCF(6๐ฅ๐ฅ4(๐ฅ๐ฅ + 1)3, 3๐ฅ๐ฅ3(๐ฅ๐ฅ + 1), 9๐ฅ๐ฅ(๐ฅ๐ฅ + 1)2) = ๐๐๐๐(๐๐ + ๐๐)
b. Since ๐ฅ๐ฅ โ ๐ฅ๐ฅ is opposite to ๐ฅ๐ฅ โ ๐ฅ๐ฅ, then ๐ฅ๐ฅ โ ๐ฅ๐ฅ can be written as โ(๐ฅ๐ฅ โ ๐ฅ๐ฅ). So 4, ๐๐, and (๐ฅ๐ฅ โ ๐ฅ๐ฅ) is common for both expressions. Thus,
GCF๏ฟฝ4๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ), 8๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ)๏ฟฝ = ๐๐๐๐(๐๐ โ ๐๐)
Note: The Greatest Common Factor is unique up to the sign. Notice that in the above example, we could write ๐ฅ๐ฅ โ ๐ฅ๐ฅ as โ(๐ฅ๐ฅ โ ๐ฅ๐ฅ) and choose the GCF to be 4๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ).
c. The terms ๐๐๐๐2, ๐๐2๐๐, ๐๐, and ๐๐ have no common factor other than 1, so
GCF(๐๐๐๐2, ๐๐2๐๐, ๐๐, ๐๐) = ๐๐
Solution
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d. The lowest power out of ๐ฅ๐ฅโ1 and ๐ฅ๐ฅโ2 is ๐ฅ๐ฅโ2, and the lowest power out of ๐ฅ๐ฅโ3 and ๐ฅ๐ฅโ2 is ๐ฅ๐ฅโ3. Therefore,
GCF(3๐ฅ๐ฅโ1๐ฅ๐ฅโ3, ๐ฅ๐ฅโ2๐ฅ๐ฅโ2๐ง๐ง) = ๐๐โ๐๐๐๐โ๐๐
Factoring out the Greatest Common Factor Factor each expression by taking the greatest common factor out. Simplify the factors, if possible.
a. 54๐ฅ๐ฅ2๐ฅ๐ฅ2 + 60๐ฅ๐ฅ๐ฅ๐ฅ3 b. ๐๐๐๐ โ ๐๐2๐๐(๐๐ โ 1)
c. โ๐ฅ๐ฅ(๐ฅ๐ฅ โ 5) + ๐ฅ๐ฅ2(5 โ ๐ฅ๐ฅ) โ (๐ฅ๐ฅ โ 5)2 d. ๐ฅ๐ฅโ1 + 2๐ฅ๐ฅโ2 โ ๐ฅ๐ฅโ3 a. To find the greatest common factor of 54 and 60, we can use the method of dividing
by any common factor, as presented below.
2 54, 603 27, 30 9, 10
So, GCF(54, 60) = 2 โ 3 = 6.
Since GCF(54๐ฅ๐ฅ2๐ฅ๐ฅ2, 60๐ฅ๐ฅ๐ฅ๐ฅ3) = 6๐ฅ๐ฅ๐ฅ๐ฅ2, we factor the 6๐ฅ๐ฅ๐ฅ๐ฅ2 out by dividing each term of the polynomial 54๐ฅ๐ฅ2๐ฅ๐ฅ2 + 60๐ฅ๐ฅ๐ฅ๐ฅ3 by 6๐ฅ๐ฅ๐ฅ๐ฅ2, as below.
54๐ฅ๐ฅ2๐ฅ๐ฅ2 + 60๐ฅ๐ฅ๐ฅ๐ฅ3
= ๐๐๐๐๐๐๐๐(๐๐๐๐+ ๐๐๐๐๐๐) Note: Since factoring is the reverse process of multiplication, it can be checked by finding the product of the factors. If the product gives us the original polynomial, the factorization is correct.
b. First, notice that the polynomial has two terms, ๐๐๐๐ and โ๐๐2๐๐(๐๐ โ 1). The greatest common factor for these two terms is ๐๐๐๐, so we have
๐๐๐๐ โ ๐๐2๐๐(๐๐ โ 1) = ๐๐๐๐๏ฟฝ๐๐ โ ๐๐(๐๐ โ 1)๏ฟฝ
= ๐๐๐๐(1 โ ๐๐2 + ๐๐) = ๐๐๐๐(โ๐๐2 + ๐๐ + 1)
= โ๐๐๐๐๏ฟฝ๐๐๐๐ โ ๐๐ โ ๐๐๏ฟฝ
Solution
no more common factors
for 9 and 10
all common factors are listed in this column
54๐ฅ๐ฅ2๐ฅ๐ฅ2
6๐ฅ๐ฅ๐ฅ๐ฅ2= 9๐ฅ๐ฅ
60๐ฅ๐ฅ๐ฅ๐ฅ3
6๐ฅ๐ฅ๐ฅ๐ฅ2= 10๐ฅ๐ฅ
remember to leave 1 for the first term
simplify and arrange in decreasing powers
take the โโโ out
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Factoring
Note: Both factorizations, ๐๐๐๐(โ๐๐2 + ๐๐ + 1) and โ๐๐๐๐(๐๐2 โ ๐๐ โ 1) are correct. However, we customarily leave the polynomial in the bracket with a positive leading coefficient.
c. Observe that if we write the middle term ๐ฅ๐ฅ2(5 โ ๐ฅ๐ฅ) as โ๐ฅ๐ฅ2(๐ฅ๐ฅ โ 5) by factoring the
negative out of the (5 โ ๐ฅ๐ฅ), then (5 โ ๐ฅ๐ฅ) is the common factor of all the terms of the equivalent polynomial
โ๐ฅ๐ฅ(๐ฅ๐ฅ โ 5) โ ๐ฅ๐ฅ2(๐ฅ๐ฅ โ 5) โ (๐ฅ๐ฅ โ 5)2.
Then notice that if we take โ(๐ฅ๐ฅ โ 5) as the GCF, then the leading term of the remaining polynomial will be positive. So, we factor
โ๐ฅ๐ฅ(๐ฅ๐ฅ โ 5) + ๐ฅ๐ฅ2(5โ ๐ฅ๐ฅ) โ (๐ฅ๐ฅ โ 5)2
= โ๐ฅ๐ฅ(๐ฅ๐ฅ โ 5) โ ๐ฅ๐ฅ2(๐ฅ๐ฅ โ 5) โ (๐ฅ๐ฅ โ 5)2
= โ(๐ฅ๐ฅ โ 5)๏ฟฝ๐ฅ๐ฅ + ๐ฅ๐ฅ2 + (๐ฅ๐ฅ โ 5)๏ฟฝ
= โ(๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ โ ๐๐๏ฟฝ
d. The GCF(๐ฅ๐ฅโ1, 2๐ฅ๐ฅโ2, โ๐ฅ๐ฅโ3) = ๐ฅ๐ฅโ3, as โ3 is the lowest exponent of the common factor ๐ฅ๐ฅ. So, we factor out ๐ฅ๐ฅโ3 as below.
๐ฅ๐ฅโ1 + 2๐ฅ๐ฅโ2 โ ๐ฅ๐ฅโ3
= ๐๐โ๐๐ ๏ฟฝ๐๐๐๐ + ๐๐๐๐ โ ๐๐๏ฟฝ
To check if the factorization is correct, we multiply
๐ฅ๐ฅโ3 (๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 1)
= ๐ฅ๐ฅโ3๐ฅ๐ฅ2 + 2๐ฅ๐ฅโ3๐ฅ๐ฅ โ 1๐ฅ๐ฅโ3
= ๐ฅ๐ฅโ1 + 2๐ฅ๐ฅโ2 โ ๐ฅ๐ฅโ3
Since the product gives us the original polynomial, the factorization is correct.
Factoring by Grouping
Consider the polynomial ๐ฅ๐ฅ2 + ๐ฅ๐ฅ + ๐ฅ๐ฅ๐ฅ๐ฅ + ๐ฅ๐ฅ. It consists of four terms that do not have any common factors. Yet, it can still be factored if we group the first two and the last two terms. The first group of two terms contains the common factor of ๐ฅ๐ฅ and the second group of two terms contains the common factor of ๐ฅ๐ฅ. Observe what happens when we factor each group.
๐ฅ๐ฅ2 + ๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ
+ ๐ฅ๐ฅ๐ฅ๐ฅ + ๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ
= ๐ฅ๐ฅ(๐ฅ๐ฅ + 1) + ๐ฅ๐ฅ(๐ฅ๐ฅ + 1)
= (๐ฅ๐ฅ + 1)(๐ฅ๐ฅ + ๐ฅ๐ฅ)
When referring to a common factor, we
have in mind a common factor other
than 1.
the exponent 2 is found by subtracting โ3 from โ1
the exponent 1 is found by subtracting โ3 from โ2
add exponents
now (๐ฅ๐ฅ + 1) is the
common factor of the entire polynomial
simplify and arrange in decreasing powers
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This method is called factoring by grouping, in particular, two-by-two grouping.
Warning: After factoring each group, make sure to write the โ+โ or โโโ between the terms. Failing to write these signs leads to the false impression that the polynomial is already factored. For example, if in the second line of the above calculations we would fail to write the middle โ+โ, the expression would look like a product ๐ฅ๐ฅ(๐ฅ๐ฅ + 1) ๐ฅ๐ฅ(๐ฅ๐ฅ + 1), which is not the case. Also, since the expression ๐ฅ๐ฅ(๐ฅ๐ฅ + 1) + ๐ฅ๐ฅ(๐ฅ๐ฅ + 1) is a sum, not a product, we should not stop at this step. We need to factor out the common bracket (๐ฅ๐ฅ + 1) to leave it as a product.
A two-by-two grouping leads to a factorization only if the binomials, after factoring out the common factors in each group, are the same. Sometimes a rearrangement of terms is necessary to achieve this goal. For example, the attempt to factor ๐ฅ๐ฅ3 โ 15 + 5๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ by grouping the first and the last two terms,
๐ฅ๐ฅ3 โ 15๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ 5๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
= (๐ฅ๐ฅ3 โ 15) + ๐ฅ๐ฅ(5๐ฅ๐ฅ โ 3)
does not lead us to a common binomial that could be factored out.
However, rearranging terms allows us to factor the original polynomial in the following ways:
๐ฅ๐ฅ3 โ 15 + 5๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ or ๐ฅ๐ฅ3 โ 15 + 5๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ
= ๐ฅ๐ฅ3 + 5๐ฅ๐ฅ2๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ โ3๐ฅ๐ฅ โ 15๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
= ๐ฅ๐ฅ3 โ 3๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ 5๐ฅ๐ฅ2 โ 15๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
= ๐ฅ๐ฅ2(๐ฅ๐ฅ + 5) โ 3(๐ฅ๐ฅ + 5) = ๐ฅ๐ฅ(๐ฅ๐ฅ2 โ 3) + 5(๐ฅ๐ฅ2 โ 3)
= (๐ฅ๐ฅ + 5)(๐ฅ๐ฅ2 โ 3) = (๐ฅ๐ฅ2 โ 3)(๐ฅ๐ฅ + 5) Factoring by grouping applies to polynomials with more than three terms. However, not all such polynomials can be factored by grouping. For example, if we attempt to factor ๐ฅ๐ฅ3 +๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 2 by grouping, we obtain
๐ฅ๐ฅ3 + ๐ฅ๐ฅ2๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ 2๐ฅ๐ฅ โ 2๏ฟฝ๏ฟฝ๏ฟฝ
= ๐ฅ๐ฅ2(๐ฅ๐ฅ + 1) + 2(๐ฅ๐ฅ โ 1).
Unfortunately, the expressions ๐ฅ๐ฅ + 1 and ๐ฅ๐ฅ โ 1 are not the same, so there is no common factor to factor out. One can also check that no other rearrangments of terms allows us for factoring out a common binomial. So, this polynomial cannot be factored by grouping.
Factoring by Grouping
Factor each polynomial by grouping, if possible. Remember to check for the GCF first.
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section F1 | 220
Factoring
a. 2๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 3 b. 5๐ฅ๐ฅ โ 5๐ฅ๐ฅ โ ๐๐๐ฅ๐ฅ + ๐๐๐ฅ๐ฅ c. 2๐ฅ๐ฅ2๐ฅ๐ฅ โ 8 โ 2๐ฅ๐ฅ2 + 8๐ฅ๐ฅ d. ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + ๐ฅ๐ฅ + 1 a. Since there is no common factor for all four terms, we will attempt the two-by-two
grouping method. 2๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ2๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
+ ๐ฅ๐ฅ โ 3๏ฟฝ๏ฟฝ๏ฟฝ
= 2๐ฅ๐ฅ2(๐ฅ๐ฅ โ 3) + 1(๐ฅ๐ฅ โ 3)
= (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐๐๐ + ๐๐๏ฟฝ
b. As before, there is no common factor for all four terms. The two-by-two grouping method works only if the remaining binomials after factoring each group are exactly the same. We can achieve this goal by factoring โ๐๐ , rather than ๐๐, out of the last two terms. So,
5๐ฅ๐ฅ โ 5๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
โ๐๐๐ฅ๐ฅ + ๐๐๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
= 5(๐ฅ๐ฅ โ ๐ฅ๐ฅ) โ ๐๐(๐ฅ๐ฅ โ ๐ฅ๐ฅ)
= (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐๐๐ + ๐๐๏ฟฝ
c. Notice that 2 is the GCF of all terms, so we factor it out first.
2๐ฅ๐ฅ2๐ฅ๐ฅ โ 8 โ 2๐ฅ๐ฅ2 + 8๐ฅ๐ฅ
= 2(๐ฅ๐ฅ2๐ฅ๐ฅ โ 4 โ ๐ฅ๐ฅ2 + 4๐ฅ๐ฅ)
Then, observe that grouping the first and last two terms of the remaining polynomial does not help, as the two groups do not have any common factors. However, exchanging for example the second with the fourth term will help, as shown below.
= 2(๐ฅ๐ฅ2๐ฅ๐ฅ + 4๐ฅ๐ฅ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ
โ๐ฅ๐ฅ2 โ 4๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ)
= 2[๐ฅ๐ฅ(๐ฅ๐ฅ2 + 4) โ (๐ฅ๐ฅ2 + 4)]
= ๐๐๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ(๐๐ โ ๐๐)
d. The polynomial ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + ๐ฅ๐ฅ + 1 does not have any common factors for all four terms. Also, only the first two terms have a common factor. Unfortunately, when attempting to factor using the two-by-two grouping method, we obtain
๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + ๐ฅ๐ฅ + 1
= ๐ฅ๐ฅ(๐ฅ๐ฅ โ 1) + (๐ฅ๐ฅ + 1),
which cannot be factored, as the expressions ๐ฅ๐ฅ โ 1 and ๐ฅ๐ฅ + 1 are different.
One can also check that no other arrangement of terms allows for factoring of this polynomial by grouping. So, this polynomial cannot be factored by grouping.
Solution
write the 1 for the second term
reverse signs when โpullingโ a โโโ out
reverse signs when โpullingโ a โโโ out
the square bracket is
essential here because of the factor of 2
now, there is no need for the square bracket as multiplication is associative
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section F1 | 221
Greatest Common Factor and Factoring by Grouping
Factoring in Solving Formulas
Solve ๐๐๐๐ = 3๐๐ + 5 for ๐๐.
First, we move the terms containing the variable ๐๐ to one side of the equation,
๐๐๐๐ = 3๐๐ + 5 ๐๐๐๐ โ 3๐๐ = 5,
and then factor ๐๐ out ๐๐(๐๐ โ 3) = 5.
So, after dividing by ๐๐ โ 3, we obtain ๐๐ = ๐๐๐๐โ๐๐
.
F.1 Exercises
Vocabulary Check Complete each blank with the most appropriate term or phrase from the given list: common factor, distributive, grouping, prime, product.
1. To factor a polynomial means to write it as a ____________ of simpler polynomials.
2. The greatest ______________ _____________ of two or more terms is the product of the least powers of each type of common factor out of all the terms.
3. To factor out the GCF, we reverse the ______________ property of multiplication.
4. A polynomial with four terms having no common factors can be still factored by _____________ its terms.
5. A ___________ polynomial, other than a monomial, cannot be factored into two polynomials, both different that 1 or โ1.
Concept Check True or false.
6. The polynomial 6๐ฅ๐ฅ + 8๐ฅ๐ฅ is prime.
7. The factorization 12๐ฅ๐ฅ โ 3
4๐ฅ๐ฅ = 1
4(2๐ฅ๐ฅ โ 3๐ฅ๐ฅ) is essential to be complete.
8. The GCF of the terms of the polynomial 3(๐ฅ๐ฅ โ 2) + ๐ฅ๐ฅ(2 โ ๐ฅ๐ฅ) is (๐ฅ๐ฅ โ 2)(2 โ ๐ฅ๐ฅ). Concept Check Find the GCF with a positive coefficient for the given expressions.
9. 8๐ฅ๐ฅ๐ฅ๐ฅ, 10๐ฅ๐ฅ๐ง๐ง, โ14๐ฅ๐ฅ๐ฅ๐ฅ 10. 21๐๐3๐๐6, โ35๐๐7๐๐5, 28๐๐5๐๐8
11. 4๐ฅ๐ฅ(๐ฅ๐ฅ โ 1), 3๐ฅ๐ฅ2(๐ฅ๐ฅ โ 1) 12. โ๐ฅ๐ฅ(๐ฅ๐ฅ โ 3)2, ๐ฅ๐ฅ2(๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 2)
13. 9(๐๐ โ 5), 12(5โ ๐๐) 14. (๐ฅ๐ฅ โ 2๐ฅ๐ฅ)(๐ฅ๐ฅ โ 1), (2๐ฅ๐ฅ โ ๐ฅ๐ฅ)(๐ฅ๐ฅ + 1)
Solution
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section F1 | 222
Factoring
15. โ3๐ฅ๐ฅโ2๐ฅ๐ฅโ3, 6๐ฅ๐ฅโ3๐ฅ๐ฅโ5 16. ๐ฅ๐ฅโ2(๐ฅ๐ฅ + 2)โ2, โ๐ฅ๐ฅโ4(๐ฅ๐ฅ + 2)โ1 Factor out the greatest common factor. Leave the remaining polynomial with a positive leading coeficient. Simplify the factors, if possible.
17. 9๐ฅ๐ฅ2 โ 81๐ฅ๐ฅ 18. 8๐๐3 + 24๐๐ 19. 6๐๐3 โ 3๐๐2 โ 9๐๐4
20. 6๐๐3 โ 36๐๐4 + 18๐๐2 21. โ10๐๐2๐ ๐ 2 + 15๐๐4๐ ๐ 2 22. 5๐ฅ๐ฅ2๐ฅ๐ฅ3 โ 10๐ฅ๐ฅ3๐ฅ๐ฅ2
23. ๐๐(๐ฅ๐ฅ โ 2) + ๐๐(๐ฅ๐ฅ โ 2) 24. ๐๐(๐ฅ๐ฅ2 โ 3) โ 2(๐ฅ๐ฅ2 โ 3)
25. (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 3) + (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 5) 26. (๐๐ โ 2)(๐๐ + 3) + (๐๐ โ 2)(๐๐ โ 3)
27. ๐ฅ๐ฅ(๐ฅ๐ฅ โ 1) + 5(1 โ ๐ฅ๐ฅ) 28. (4๐ฅ๐ฅ โ ๐ฅ๐ฅ) โ 4๐ฅ๐ฅ(๐ฅ๐ฅ โ 4๐ฅ๐ฅ)
29. 4(3 โ ๐ฅ๐ฅ)2 โ (3 โ ๐ฅ๐ฅ)3 + 3(3 โ ๐ฅ๐ฅ) 30. 2(๐๐ โ 3) + 4(๐๐ โ 3)2 โ (๐๐ โ 3)3 Factor out the least power of each variable.
31. 3๐ฅ๐ฅโ3 + ๐ฅ๐ฅโ2 32. ๐๐โ2 + 2๐๐โ4 33. ๐ฅ๐ฅโ4 โ 2๐ฅ๐ฅโ3 + 7๐ฅ๐ฅโ2
34. 3๐๐โ5 + ๐๐โ3 โ 2๐๐โ2 35. 3๐ฅ๐ฅโ3๐ฅ๐ฅ โ ๐ฅ๐ฅโ2๐ฅ๐ฅ2 36. โ5๐ฅ๐ฅโ2๐ฅ๐ฅโ3 + 2๐ฅ๐ฅโ1๐ฅ๐ฅโ2 Factor by grouping, if possible.
37. 20 + 5๐ฅ๐ฅ + 12๐ฅ๐ฅ + 3๐ฅ๐ฅ๐ฅ๐ฅ 38. 2๐๐3 + ๐๐2 โ 14๐๐ โ 7 39. ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐ โ ๐๐๐๐
40. 2๐ฅ๐ฅ๐ฅ๐ฅ โ ๐ฅ๐ฅ2๐ฅ๐ฅ + 6 โ 3๐ฅ๐ฅ 41. 3๐ฅ๐ฅ2 + 4๐ฅ๐ฅ๐ฅ๐ฅ โ 6๐ฅ๐ฅ๐ฅ๐ฅ โ 8๐ฅ๐ฅ2 42. ๐ฅ๐ฅ3 โ ๐ฅ๐ฅ๐ฅ๐ฅ + ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ2๐ฅ๐ฅ
43. 3๐๐2 + 9๐๐๐๐ โ ๐๐๐๐ โ 3๐๐2 44. 3๐ฅ๐ฅ2 โ ๐ฅ๐ฅ2๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ง๐ง2 + 3๐ง๐ง2 45. 2๐ฅ๐ฅ3 โ ๐ฅ๐ฅ2 + 4๐ฅ๐ฅ โ 2
46. ๐ฅ๐ฅ2๐ฅ๐ฅ2 + ๐๐๐๐ โ ๐๐๐ฅ๐ฅ2 โ ๐๐๐ฅ๐ฅ2 47. ๐ฅ๐ฅ๐ฅ๐ฅ + ๐๐๐๐ + ๐๐๐ฅ๐ฅ + ๐๐๐ฅ๐ฅ 48. ๐ฅ๐ฅ2๐ฅ๐ฅ โ ๐ฅ๐ฅ๐ฅ๐ฅ + ๐ฅ๐ฅ + ๐ฅ๐ฅ
49. ๐ฅ๐ฅ๐ฅ๐ฅ โ 6๐ฅ๐ฅ + 3๐ฅ๐ฅ โ 18 50. ๐ฅ๐ฅ๐๐๐ฅ๐ฅ โ 3๐ฅ๐ฅ๐๐ + ๐ฅ๐ฅ โ 5 51. ๐๐๐๐๐ฅ๐ฅ๐๐ + 2๐๐๐๐ + ๐ฅ๐ฅ๐๐ + 2
Factor completely. Remember to check for the GCF first.
52. 5๐ฅ๐ฅ โ 5๐๐๐ฅ๐ฅ + 5๐๐๐๐๐๐ โ 5๐๐๐๐ 53. 6๐๐๐ ๐ โ 14๐ ๐ + 6๐๐ โ 14
54. ๐ฅ๐ฅ4(๐ฅ๐ฅ โ 1) + ๐ฅ๐ฅ3(๐ฅ๐ฅ โ 1) โ ๐ฅ๐ฅ2 + ๐ฅ๐ฅ 55. ๐ฅ๐ฅ3(๐ฅ๐ฅ โ 2)2 + 2๐ฅ๐ฅ2(๐ฅ๐ฅ โ 2) โ (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 2) Discussion Point
56. One of possible factorizations of the polynomial 4๐ฅ๐ฅ2๐ฅ๐ฅ5 โ 8๐ฅ๐ฅ๐ฅ๐ฅ3 is 2๐ฅ๐ฅ๐ฅ๐ฅ3(2๐ฅ๐ฅ๐ฅ๐ฅ2 โ 4). Is this a complete factorization?
Use factoring the GCF strategy to solve each formula for the indicated variable.
57. ๐ด๐ด = ๐ท๐ท + ๐ท๐ท๐๐, for ๐ท๐ท 58. ๐๐ = 12๐๐๐๐ + 1
2๐๐๐๐, for ๐๐
59. 2๐๐ + ๐๐ = ๐๐๐๐, for ๐๐ 60. ๐ค๐ค๐๐ = 3๐๐ โ ๐ฅ๐ฅ, for ๐๐
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section F1 | 223
Greatest Common Factor and Factoring by Grouping
Analytic Skills Write the area of each shaded region in factored form. 61. 62.
63. 64.
4๐ฅ๐ฅ
๐ฅ๐ฅ ๐ฅ๐ฅ
๐๐ ๐ ๐ ๐๐
๐๐
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section F2 | 224
Factoring
F.2 Factoring Trinomials
In this section, we discuss factoring trinomials. We start with factoring quadratic trinomials of the form ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐, then quadratic trinomials of the form ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐, where ๐๐ โ 1, and finally trinomials reducible to quadratic by means of substitution.
Factorization of Quadratic Trinomials ๐๐๐๐ + ๐๐๐๐ + ๐๐
Factorization of a quadratic trinomial ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ is the reverse process of the FOIL method of multiplying two linear binomials. Observe that
(๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ + ๐๐) = ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐๐ฅ๐ฅ + ๐๐๐๐ = ๐ฅ๐ฅ2 + (๐๐ + ๐๐)๐ฅ๐ฅ + ๐๐๐๐
So, to reverse this multiplication, we look for two numbers ๐๐ and ๐๐, such that the product ๐๐๐๐ equals to the free term ๐๐ and the sum ๐๐ + ๐๐ equals to the middle coefficient ๐๐ of the trinomial.
๐ฅ๐ฅ2 + ๐๐โ(๐๐+๐๐)
๐ฅ๐ฅ + ๐๐โ๐๐๐๐
= (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ + ๐๐)
For example, to factor ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 6, we think of two integers that multiply to 6 and add to 5. Such integers are 2 and 3, so ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 6 = (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ + 3). Since multiplication is commutative, the order of these factors is not important.
This could also be illustrated geometrically, using algebra tiles.
The area of a square with the side length ๐ฅ๐ฅ is equal to ๐ฅ๐ฅ2. The area of a rectangle with the dimensions ๐ฅ๐ฅ by 1 is equal to ๐ฅ๐ฅ, and the area of a unit square is equal to 1. So, the trinomial ๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 6 can be represented as
To factor this trinomial, we would like to rearrange these tiles to fulfill a rectangle.
The area of such rectangle can be represented as the product of its length, (๐ฅ๐ฅ + 3), and width, (๐ฅ๐ฅ + 2) which becomes the factorization of the original trinomial. In the trinomial examined above, the signs of the middle and the last terms are both positive. To analyse how different signs of these terms influence the signs used in the factors, observe the next three examples.
๐ฅ๐ฅ2 ๐ฅ๐ฅ
1
๐ฅ๐ฅ2 ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ฅ๐ฅ
1 1 1 1 1 1
๐ฅ๐ฅ2 ๐ฅ๐ฅ ๐ฅ๐ฅ ๐ฅ๐ฅ
๐ฅ๐ฅ ๐ฅ๐ฅ 1 1
1 1 1 1
๐ฅ๐ฅ + 3 ๏ฟฝโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ๏ฟฝ
๏ฟฝโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏโฏ๏ฟฝ
๐ฅ๐ฅ +
2
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section F2 | 225
Factoring Trinomials
To factor ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + 6, we look for two integers that multiply to 6 and add to โ5. Such integers are โ2 and โ3, so ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + 6 = (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ โ 3). To factor ๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6, we look for two integers that multiply to โ6 and add to 1. Such integers are โ2 and 3, so ๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 3). To factor ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 6, we look for two integers that multiply to โ6 and add to โ1. Such integers are 2 and โ3, so ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 6 = (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 3). Observation: The positive constant ๐๐ in a trinomial ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ tells us that the integers ๐๐ and ๐๐ in the factorization (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ + ๐๐) are both of the same sign and their sum is the middle coefficient ๐๐. In addition, if ๐๐ is positive, both ๐๐ and ๐๐ are positive, and if ๐๐ is negative, both ๐๐ and ๐๐ are negative.
The negative constant ๐๐ in a trinomial ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ tells us that the integers ๐๐ and ๐๐ in the factorization (๐ฅ๐ฅ + ๐๐)(๐ฅ๐ฅ + ๐๐) are of different signs and a difference of their absolute values is the middle coefficient ๐๐. In addition, the integer whose absolute value is larger takes the sign of the middle coefficient ๐๐. These observations are summarized in the following Table of Signs.
Assume that |๐๐| โฅ |๐๐|. sum ๐๐ product ๐๐ ๐๐ ๐๐ comments
+ + + + ๐๐ is the sum of ๐๐ and ๐๐ โ + โ โ ๐๐ is the sum of ๐๐ and ๐๐ + โ + โ ๐๐ is the difference |๐๐| โ |๐๐| โ โ โ + ๐๐ is the difference |๐๐| โ |๐๐|
Factoring Trinomials with the Leading Coefficient Equal to 1
Factor each trinomial, if possible.
a. ๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ + 24 b. ๐ฅ๐ฅ2 + 9๐ฅ๐ฅ โ 36 c. ๐ฅ๐ฅ2 โ 39๐ฅ๐ฅ๐ฅ๐ฅ โ 40๐ฅ๐ฅ2 d. ๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 9 a. To factor the trinomial ๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ + 24, we look for two integers with a product of 24
and a sum of โ10. The two integers are fairly easy to guess, โ4 and โ6. However, if one wishes to follow a more methodical way of finding these numbers, one can list the possible two-number factorizations of 24 and observe the sums of these numbers.
product = ๐๐๐๐ (pairs of factors of 24)
sum = โ๐๐๐๐ (sum of factors)
๐๐ โ ๐๐๐๐ 25 ๐๐ โ ๐๐๐๐ 14 ๐๐ โ ๐๐ 11 ๐๐ โ ๐๐ 10
Solution
๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ๐ฉ!
For simplicity, the table doesnโt include signs of the
integers. The signs are determined according to
the Table of Signs.
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section F2 | 226
Factoring
Since the product is positive and the sum is negative, both integers must be negative. So, we take โ4 and โ6.
Thus, ๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ + 24 = (๐๐ โ ๐๐)(๐๐ โ ๐๐). The reader is encouraged to check this factorization by multiplying the obtained binomials.
b. To factor the trinomial ๐ฅ๐ฅ2 + 9๐ฅ๐ฅ โ 36, we look for two integers with a product of โ36
and a sum of 9. So, let us list the possible factorizations of 36 into two numbers and observe the differences of these numbers.
product = โ๐๐๐๐ (pairs of factors of 36)
sum = ๐๐ (difference of factors)
๐๐ โ ๐๐๐๐ 35 ๐๐ โ ๐๐๐๐ 16 ๐๐ โ ๐๐๐๐ 9 ๐๐ โ ๐๐ 5 ๐๐ โ ๐๐ 0
Since the product is negative and the sum is positive, the integers are of different signs
and the one with the larger absolute value assumes the sign of the sum, which is positive. So, we take 12 and โ3.
Thus, ๐ฅ๐ฅ2 + 9๐ฅ๐ฅ โ 36 = (๐๐ + ๐๐๐๐)(๐๐ โ ๐๐). Again, the reader is encouraged to check this factorization by mltiplying the obtained binomials.
c. To factor the trinomial ๐ฅ๐ฅ2 โ 39๐ฅ๐ฅ๐ฅ๐ฅ โ 40๐ฅ๐ฅ2, we look for two binomials of the form
(๐ฅ๐ฅ+ ?๐ฅ๐ฅ)(๐ฅ๐ฅ+ ?๐ฅ๐ฅ) where the question marks are two integers with a product of โ40 and a sum of 39. Since the two integers are of different signs and the absolute values of these integers differ by 39, the two integers must be โ40 and 1.
Therefore, ๐ฅ๐ฅ2 โ 39๐ฅ๐ฅ๐ฅ๐ฅ โ 40๐ฅ๐ฅ2 = (๐๐ โ ๐๐๐๐๐๐)(๐๐ + ๐๐). Suggestion: Create a table of pairs of factors only if guessing the two integers with the given product and sum becomes too difficult. d. When attempting to factor the trinomial ๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 9, we look for a pair of integers
that would multiply to 9 and add to 7. There are only two possible factorizations of 9: 9 โ 1 and 3 โ 3. However, neither of the sums, 9 + 1 or 3 + 3, are equal to 7. So, there is no possible way of factoring ๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 9 into two linear binomials with integral coefficients. Therefore, if we admit only integral coefficients, this polynomial is not factorable.
Factorization of Quadratic Trinomials ๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐ with ๐๐ โ 0
Before discussing factoring quadratic trinomials with a leading coefficient different than 1, let us observe the multiplication process of two linear binomials with integral coefficients.
(๐๐๐ฅ๐ฅ + ๐๐)(๐ฉ๐ฉ๐ฅ๐ฅ + ๐๐) = ๐๐๐๐๐ฅ๐ฅ2 + ๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐ = ๐๐โ๐๐๐ฉ๐ฉ
๐ฅ๐ฅ2 + ๐๐โ(๐๐๐๐+๐ฉ๐ฉ๐๐)
๐ฅ๐ฅ + ๐๐โ๐๐๐๐
This row contains the solution, so there is no need to list any of the
subsequent rows.
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section F2 | 227
Factoring Trinomials
To reverse this process, notice that this time, we are looking for four integers ๐๐, ๐๐, ๐๐, and ๐๐ that satisfy the conditions
๐๐๐๐ = ๐๐, ๐๐๐๐ = ๐๐, ๐๐๐๐ + ๐๐๐๐ = ๐๐,
where ๐๐, ๐๐, ๐๐ are the coefficients of the quadratic trinomial that needs to be factored. This produces a lot more possibilities to consider than in the guessing method used in the case of the leading coefficient equal to 1. However, if at least one of the outside coefficients, ๐๐ or ๐๐, are prime, the guessing method still works reasonably well.
For example, consider 2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6. Since the coefficient ๐๐ = 2 = ๐๐๐๐ is a prime number, there is only one factorization of ๐๐, which is 1 โ 2. So, we can assume that ๐๐ = 2 and ๐๐ =1. Therefore,
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ ยฑ |๐๐|)(๐ฅ๐ฅ โ |๐๐|)
Since the constant term ๐๐ = โ6 = ๐๐๐๐ is negative, the binomial factors have different signs in the middle. Also, since ๐๐๐๐ is negative, we search for such ๐๐ and ๐๐ that the inside and outside products differ by the middle term ๐๐ = ๐ฅ๐ฅ, up to its sign. The only factorizations of 6 are 1 โ 6 and 2 โ 3. So we try
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ ยฑ 1)(๐ฅ๐ฅ โ 6)
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ ยฑ 6)(๐ฅ๐ฅ โ 1)
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ ยฑ 2)(๐ฅ๐ฅ โ 3)
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ ยฑ 3)(๐ฅ๐ฅ โ 2)
Then, since the difference between the inner and outer products should be positive, the larger product must be positive and the smaller product must be negative. So, we distribute the signs as below.
2๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 6 = (2๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 2)
In the end, it is a good idea to multiply the product to check if it results in the original polynomial. We leave this task to the reader. What if the outside coefficients of the quadratic trinomial are both composite? Checking all possible distributions of coefficients ๐๐, ๐๐, ๐๐, and ๐๐ might be too cumbersome. Luckily, there is another method of factoring, called decomposition.
๐ฅ๐ฅ
12๐ฅ๐ฅ differs by 11๐ฅ๐ฅ โ too much
6๐ฅ๐ฅ 2๐ฅ๐ฅ
differs by 4๐ฅ๐ฅ โ still too much
2๐ฅ๐ฅ 6๐ฅ๐ฅ
differs by 4๐ฅ๐ฅ โ still too much
3๐ฅ๐ฅ 4๐ฅ๐ฅ
differs by ๐ฅ๐ฅ โ perfect!
โ3๐ฅ๐ฅ 4๐ฅ๐ฅ
Observe that these two trials can be disregarded at once as 2 is not a common factor
of all the terms of the trinomial, while it is a
common factor of the terms of one of the binomials.
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section F2 | 228
Factoring
The decomposition method is based on the reverse FOIL process. Suppose the polynomial 6๐ฅ๐ฅ2 + 19๐ฅ๐ฅ + 15 factors into (๐๐๐ฅ๐ฅ + ๐๐)(๐๐๐ฅ๐ฅ + ๐๐). Observe that the FOIL multiplication of these two binomials results in the four term polynomial,
๐๐๐๐๐ฅ๐ฅ2 + ๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐๐ฅ๐ฅ + ๐๐๐๐,
which after combining the two middle terms gives us the original trinomial. So, reversing these steps would lead us to the factored form of 6๐ฅ๐ฅ2 + 19๐ฅ๐ฅ + 15.
To reverse the FOIL process, we would like to:
โข Express the middle term, 19๐ฅ๐ฅ, as a sum of two terms, ๐๐๐๐๐ฅ๐ฅ and ๐๐๐๐๐ฅ๐ฅ, such that the product of their coefficients, ๐๐๐๐๐๐๐๐, is equal to the product of the outside coefficients ๐๐๐๐ = 6 โ 15 = 90.
โข Then, factor the four-term polynomial by grouping.
Thus, we are looking for two integers with the product of 90 and the sum of 19. One can check that 9 and 10 satisfy these conditions. Therefore,
6๐ฅ๐ฅ2 + 19๐ฅ๐ฅ + 15
= 6๐ฅ๐ฅ2 + 9๐ฅ๐ฅ + 10๐ฅ๐ฅ + 15
= 3๐ฅ๐ฅ(2๐ฅ๐ฅ + 3) + 5(2๐ฅ๐ฅ + 3)
= (2๐ฅ๐ฅ + 3)(3๐ฅ๐ฅ + 5)
Factoring Trinomials with the Leading Coefficient Different than 1
Factor completely each trinomial.
a. 6๐ฅ๐ฅ3 + 14๐ฅ๐ฅ2 + 4๐ฅ๐ฅ b. โ6๐ฅ๐ฅ2 โ 10 + 19๐ฅ๐ฅ c. 18๐๐2 โ 19๐๐๐๐ โ 12๐๐2 d. 2(๐ฅ๐ฅ + 3)2 + 5(๐ฅ๐ฅ + 3) โ 12 a. First, we factor out the GCF, which is 2๐ฅ๐ฅ. This gives us
6๐ฅ๐ฅ3 + 14๐ฅ๐ฅ2 + 4๐ฅ๐ฅ = 2๐ฅ๐ฅ(3๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 2)
The outside coefficients of the remaining trinomial are prime, so we can apply the guessing method to factor it further. The first terms of the possible binomial factors must be 3๐ฅ๐ฅ and ๐ฅ๐ฅ while the last terms must be 2 and 1. Since both signs in the trinomial are positive, the signs used in the binomial factors must be both positive as well. So, we are ready to give it a try:
2๐ฅ๐ฅ(3๐ฅ๐ฅ + 2 )(๐ฅ๐ฅ + 1 ) or 2๐ฅ๐ฅ(3๐ฅ๐ฅ + 1 )(๐ฅ๐ฅ + 2 )
The first distribution of coefficients does not work as it would give us 2๐ฅ๐ฅ + 3๐ฅ๐ฅ = 5๐ฅ๐ฅ for the middle term. However, the second distribution works as ๐ฅ๐ฅ + 6๐ฅ๐ฅ = 7๐ฅ๐ฅ, which matches the middle term of the trinomial. So,
6๐ฅ๐ฅ3 + 14๐ฅ๐ฅ2 + 4๐ฅ๐ฅ = ๐๐๐๐(๐๐๐๐ + ๐๐)(๐๐ + ๐๐)
Solution
This product is often referred to as the
master product or the ๐๐๐๐-product.
2๐ฅ๐ฅ 3๐ฅ๐ฅ
๐ฅ๐ฅ 6๐ฅ๐ฅ
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section F2 | 229
Factoring Trinomials
b. Notice that the trinomial is not arranged in decreasing order of powers of ๐ฅ๐ฅ. So, first, we rearrange the last two terms to achieve the decreasing order. Also, we factor out the โ1, so that the leading term of the remaining trinomial is positive.
โ6๐ฅ๐ฅ2 โ 10 + 19๐ฅ๐ฅ = โ6๐ฅ๐ฅ2 + 19๐ฅ๐ฅ โ 10 = โ(6๐ฅ๐ฅ2 โ 19๐ฅ๐ฅ + 10)
Then, since the outside coefficients are composite, we will use the decomposition method of factoring. The ๐๐๐๐-product equals to 60 and the middle coefficient equals to โ19. So, we are looking for two integers that multiply to 60 and add to โ19. The integers that satisfy these conditions are โ15 and โ4. Hence, we factor
โ(6๐ฅ๐ฅ2 โ 19๐ฅ๐ฅ + 10)
= โ(6๐ฅ๐ฅ2 โ 15๐ฅ๐ฅ โ 4๐ฅ๐ฅ + 10)
= โ[3๐ฅ๐ฅ(2๐ฅ๐ฅ โ 5) โ 2(2๐ฅ๐ฅ โ 5)]
= โ(๐๐๐๐ โ ๐๐)(๐๐๐๐ โ ๐๐)
c. There is no common factor to take out of the polynomial 18๐๐2 โ 19๐๐๐๐ โ 12๐๐2. So, we will attempt to factor it into two binomials of the type (๐๐๐๐ ยฑ ๐๐๐๐)(๐๐๐๐ โ ๐๐๐๐), using the decomposition method. The ๐๐๐๐-product equals โ12 โ 18 = โ2 โ 2 โ 2 โ 3 โ 3 โ 3 and the middle coefficient equals โ19. To find the two integers that multiply to the ๐๐๐๐-product and add to โ19, it is convenient to group the factors of the product
2 โ 2 โ 2 โ 3 โ 3 โ 3
in such a way that the products of each group differ by 19. It turns out that grouping all the 2โs and all the 3โs satisfy this condition, as 8 and 27 differ by 19. Thus, the desired integers are โ27 and 8, as the sum of them must be โ19. So, we factor
18๐๐2 โ 19๐๐๐๐ โ 12๐๐2
= 18๐๐2 โ 27๐๐๐๐ + 8๐๐๐๐ โ 12๐๐2
= 9๐๐(2๐๐ โ 3๐๐) + 4๐๐(2๐๐ โ 3๐๐)
= (๐๐๐๐ โ ๐๐๐๐)(๐๐๐๐ + ๐๐๐๐)
d. To factor 2(๐ฅ๐ฅ + 3)2 + 5(๐ฅ๐ฅ + 3) โ 12, first, we notice that treating the group (๐ฅ๐ฅ + 3) as another variable, say ๐๐, simplify the problem to factoring the quadratic trinomial
2๐๐2 + 5๐๐ โ 12
This can be done by the guessing method. Since
2๐๐2 + 5๐๐ โ 12 = (2๐๐ โ 3)(๐๐ + 4),
then
2(๐ฅ๐ฅ + 3)2 + 5(๐ฅ๐ฅ + 3) โ 12 = [2(๐ฅ๐ฅ + 3) โ 3][(๐ฅ๐ฅ + 3) + 4]
= (2๐ฅ๐ฅ + 6 โ 3)(๐ฅ๐ฅ + 3 + 4)
= (๐๐๐๐ + ๐๐)(๐๐ + ๐๐)
remember to reverse the sign!
the square bracket is
essential because of the negative sign outside
โ3๐๐ 8๐๐
IN FORM
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section F2 | 230
Factoring
Note 1: Polynomials that can be written in the form ๐๐( )๐๐ + ๐๐( ) + ๐๐, where ๐๐ โ 0 and ( ) represents any nonconstant polynomial expression, are referred to as quadratic in form. To factor such polynomials, it is convenient to replace the expression in the bracket by a single variable, different than the original one. This was illustrated in Example 2d by substituting ๐๐ for (๐ฅ๐ฅ + 3). However, when using this substitution method, we must remember to leave the final answer in terms of the original variable. So, after factoring, we replace ๐๐ back with (๐ฅ๐ฅ + 3), and then simplify each factor.
Note 2: Some students may feel comfortable factoring polynomials quadratic in form directly, without using substitution.
Application of Factoring in Geometry Problems If the area of a trapezoid is 2๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 2 square meters and the lengths of the two parallel sides are ๐ฅ๐ฅ and ๐ฅ๐ฅ + 1 meters, then what polynomial represents the height of the trapezoid?
Using the formula for the area of a trapezoid, we write the equation 12โ(๐๐ + ๐๐) = 2๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 2
Since ๐๐ + ๐๐ = ๐ฅ๐ฅ + (๐ฅ๐ฅ + 1) = 2๐ฅ๐ฅ + 1, then we have 12โ(2๐ฅ๐ฅ + 1) = 2๐ฅ๐ฅ2 + 5๐ฅ๐ฅ + 2,
which after factoring the right-hand side gives us 12โ(2๐ฅ๐ฅ + 1) = (2๐ฅ๐ฅ + 1)(๐ฅ๐ฅ + 2).
To find โ, it is enough to divide the above equation by the common factor (2๐ฅ๐ฅ + 1) and then multiply it by 2. So,
โ = 2(๐ฅ๐ฅ + 2) = ๐๐๐๐ + ๐๐.
F.2 Exercises
Vocabulary Check Complete each blank with the most appropriate term or phrase from the given list:
decomposition, guessing, multiplication, prime, quadratic, sum, variable. 1. Any factorization can be checked by using _________________.
Solution
๐ฅ๐ฅ
๐ฅ๐ฅ + 1
โ
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section F2 | 231
Factoring Trinomials
2. To factor a quadratic trinomial with a leading coefficient equal to 1, we usually use the ______________ method.
3. To factor ๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ using the guessing method, write the trinomial as (๐ฅ๐ฅ+ ? )(๐ฅ๐ฅ+ ? ), where the question marks are two factors of ๐๐ whose _______ is ๐๐.
4. To factor a quadratic trinomial with a leading coefficient different than 1, we usually use the ______________ method. If one of the outside coefficients is a __________ number, we can still use the guessing method.
5. To factor polynomials that are _____________ in form, it is convenient to substitute a single variable (different than the original one) for the expression that appears in the first and the second power. However, the final factorization must be expressed back in the original __________.
Concept Check
6. If ๐๐๐ฅ๐ฅ2 + ๐๐๐ฅ๐ฅ + ๐๐ has no monomial factor, can either of the possible binomial factors have a monomial factor?
7. Is (2๐ฅ๐ฅ + 5)(2๐ฅ๐ฅ โ 4) a complete factorization of the polynomial 4๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 20?
8. When factoring the polynomial โ2๐ฅ๐ฅ2 โ 7๐ฅ๐ฅ + 15, students obtained the following answers: (โ2๐ฅ๐ฅ + 3)(๐ฅ๐ฅ + 5), (2๐ฅ๐ฅ โ 3)(โ๐ฅ๐ฅ โ 5), or โ(2๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 5)
Which of the above factorizations are correct?
9. Is the polynomial ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ + 2 factorable or is it prime?
Concept Check Fill in the missing factor.
10. ๐ฅ๐ฅ2 โ 4๐ฅ๐ฅ + 3 = ( )(๐ฅ๐ฅ โ 1) 11. ๐ฅ๐ฅ2 + 3๐ฅ๐ฅ โ 10 = ( )(๐ฅ๐ฅ โ 2)
12. ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ๐ฅ๐ฅ โ 20๐ฅ๐ฅ2 = (๐ฅ๐ฅ + 4๐ฅ๐ฅ)( ) 13. ๐ฅ๐ฅ2 + 12๐ฅ๐ฅ๐ฅ๐ฅ + 35๐ฅ๐ฅ2 = (๐ฅ๐ฅ + 5๐ฅ๐ฅ)( )
Factor, if possible.
14. ๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 12 15. ๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ + 35 16. ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 48
17. ๐๐2 โ ๐๐ โ 42 18. ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ + 3 19. ๐๐2 โ 12๐๐ โ 27
20. ๐๐2 โ 15๐๐ + 56 21. ๐ฅ๐ฅ2 + 3๐ฅ๐ฅ โ 28 22. 18 โ 7๐๐ โ ๐๐2
23. 20 + 8๐๐ โ ๐๐2 24. ๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ๐ฅ๐ฅ + 6๐ฅ๐ฅ2 25. ๐๐2 + 9๐๐๐๐ + 20๐๐2
Factor completely.
26. โ๐ฅ๐ฅ2 + 4๐ฅ๐ฅ + 21 27. โ๐ฅ๐ฅ2 + 14๐ฅ๐ฅ + 32 28. ๐๐4 โ 13๐๐3 โ 30๐๐2
29. ๐ฅ๐ฅ3 โ 15๐ฅ๐ฅ2 + 54๐ฅ๐ฅ 30. โ2๐ฅ๐ฅ2 + 28๐ฅ๐ฅ โ 80 31. โ3๐ฅ๐ฅ2 โ 33๐ฅ๐ฅ โ 72
32. ๐ฅ๐ฅ4๐ฅ๐ฅ + 7๐ฅ๐ฅ2๐ฅ๐ฅ โ 60๐ฅ๐ฅ 33. 24๐๐๐๐2 + 6๐๐2๐๐2 โ 3๐๐3๐๐2 34. 40 โ 35๐ก๐ก15 โ 5๐ก๐ก30
35. ๐ฅ๐ฅ4๐ฅ๐ฅ2 + 11๐ฅ๐ฅ2๐ฅ๐ฅ + 30 36. 64๐๐ โ 12๐๐5 โ ๐๐9 37. 24 โ 5๐ฅ๐ฅ๐๐ โ ๐ฅ๐ฅ2๐๐
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section F2 | 232
Factoring
Discussion Point
38. A polynomial ๐ฅ๐ฅ2 + ๐๐ ๐ฅ๐ฅ + 75 with an unknown coefficient ๐๐ by the middle term can be factored into two binomials with integral coefficients. What are the possible values of ๐๐?
Concept Check Fill in the missing factor.
39. 2๐ฅ๐ฅ2 + 7๐ฅ๐ฅ + 3 = ( )(๐ฅ๐ฅ + 3) 40. 3๐ฅ๐ฅ2 โ 10๐ฅ๐ฅ + 8 = ( )(๐ฅ๐ฅ โ 2)
41. 4๐ฅ๐ฅ2 + 8๐ฅ๐ฅ โ 5 = (2๐ฅ๐ฅ โ 1)( ) 42. 6๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 15 = (2๐ฅ๐ฅ + 3)( )
Factor completely.
43. 2๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ โ 3 44. 6๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 2 45. 4๐๐2 + 17๐๐ + 4
46. 6๐ก๐ก2 โ 13๐ก๐ก + 6 47. 10๐ฅ๐ฅ2 + 23๐ฅ๐ฅ โ 5 48. 42๐๐2 + 5๐๐ โ 25
49. 3๐๐2 โ 27๐๐ + 24 50. โ12๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ + 30 51. 6๐ฅ๐ฅ2 + 41๐ฅ๐ฅ๐ฅ๐ฅ โ 7๐ฅ๐ฅ2
52. 18๐ฅ๐ฅ2 + 27๐ฅ๐ฅ๐ฅ๐ฅ + 10๐ฅ๐ฅ2 53. 8 โ 13๐๐ + 6๐๐2 54. 15 โ 14๐๐ โ 8๐๐2
55. 30๐ฅ๐ฅ4 + 3๐ฅ๐ฅ3 โ 9๐ฅ๐ฅ2 56. 10๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ2 + 4๐ฅ๐ฅ4 57. 2๐ฅ๐ฅ6 + 7๐ฅ๐ฅ๐ฅ๐ฅ3 + 6๐ฅ๐ฅ2
58. 9๐ฅ๐ฅ2๐ฅ๐ฅ2 โ 4 + 5๐ฅ๐ฅ๐ฅ๐ฅ 59. 16๐ฅ๐ฅ2๐ฅ๐ฅ3 + 3๐ฅ๐ฅ โ 16๐ฅ๐ฅ๐ฅ๐ฅ2 60. 4๐๐4 โ 28๐๐2๐๐ + 49๐๐2
61. 4(๐ฅ๐ฅ โ 1)2 โ 12(๐ฅ๐ฅ โ 1) + 9 62. 2(๐๐ + 2)2 + 11(๐๐ + 2) + 15 63. 4๐ฅ๐ฅ2๐๐ โ 4๐ฅ๐ฅ๐๐ โ 3
Discussion Point
64. A polynomial 2๐ฅ๐ฅ2 + ๐๐ ๐ฅ๐ฅ โ 15 with an unknown coefficient ๐๐ by the middle term can be factored into two binomials with integral coefficients. What are the possible values of ?
Analytic Skills
65. If the volume of a case of apples is ๐ฅ๐ฅ3 + ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ cubic feet and the height of this box is (๐ฅ๐ฅ โ 1) feet, then what polynomial represents the area of the bottom of the case?
66. A ceremonial red carpet is rectangular in shape and covers 2๐ฅ๐ฅ2 + 11๐ฅ๐ฅ + 12 square feet. If the width of the carpet is (๐ฅ๐ฅ + 4) feet, express the length, in feet.
๐ฅ๐ฅโ
1
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section F2 | 233
Special Factoring and a General Strategy of Factoring
F.3 Special Factoring and a General Strategy of Factoring
Recall that in Section P2, we considered formulas that provide a shortcut for finding special products, such as a product of two conjugate binomials,
(๐๐ + ๐๐)(๐๐ โ ๐๐) = ๐๐๐๐ โ ๐๐๐๐,
or the perfect square of a binomial,
(๐๐ ยฑ ๐๐)2 = ๐๐๐๐ ยฑ ๐๐๐๐๐๐ + ๐๐๐๐.
Since factoring reverses the multiplication process, these formulas can be used as shortcuts in factoring binomials of the form ๐๐๐๐ โ ๐๐๐๐ (difference of squares), and trinomials of the form ๐๐๐๐ ยฑ ๐๐๐๐๐๐ + ๐๐๐๐ (perfect square). In this section, we will also introduce a formula for factoring binomials of the form ๐๐๐๐ ยฑ ๐๐๐๐ (sum or difference of cubes). These special product factoring techniques are very useful in simplifying expressions or solving equations, as they allow for more efficient algebraic manipulations.
At the end of this section, we give a summary of all the factoring strategies shown in this chapter.
Difference of Squares Out of the special factoring formulas, the easiest one to use is the difference of squares,
๐๐๐๐ โ ๐๐๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐) Figure 3.1 shows a geometric interpretation of this formula. The area of the yellow square, ๐๐2, diminished by the area of the blue square, ๐๐2, can be rearranged to a rectangle with the length of (๐๐ + ๐๐) and the width of (๐๐ โ ๐๐). To factor a difference of squares ๐๐๐๐ โ ๐๐๐๐, first, identify ๐๐ and ๐๐, which are the expressions being squared, and then, form two factors, the sum (๐๐ + ๐๐), and the difference (๐๐ โ ๐๐), as illustrated in the example below.
Factoring Differences of Squares
Factor each polynomial completely.
a. 25๐ฅ๐ฅ2 โ 1 b. 3.6๐ฅ๐ฅ4 โ 0.9๐ฅ๐ฅ6 c. ๐ฅ๐ฅ4 โ 81 d. 16 โ (๐๐ โ 2)2 a. First, we rewrite each term of 25๐ฅ๐ฅ2 โ 1 as a perfect square of an expression.
๐๐ ๐๐
25๐ฅ๐ฅ2 โ 1 = (5๐ฅ๐ฅ)2 โ 12
Then, treating 5๐ฅ๐ฅ as the ๐๐ and 1 as the ๐๐ in the difference of squares formula ๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐), we factor:
Solution
Figure 3.1
๐๐
๐๐ ๐๐ โ ๐๐ ๐๐
๐๐โ๐๐
๐๐ ๐๐ + ๐๐
๐๐2 ๐๐2
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section F3 | 234
Factoring
๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐)
25๐ฅ๐ฅ2 โ 1 = (5๐ฅ๐ฅ)2 โ 12 = (๐๐๐๐ + ๐๐)(๐๐๐๐ โ ๐๐)
b. First, we factor out 0.9 to leave the coefficients in a perfect square form. So,
3.6๐ฅ๐ฅ4 โ 0.9๐ฅ๐ฅ6 = 0.9(4๐ฅ๐ฅ4 โ ๐ฅ๐ฅ6)
Then, after writing the terms of 4๐ฅ๐ฅ4 โ ๐ฅ๐ฅ6 as perfect squares of expressions that correspond to ๐๐ and ๐๐ in the difference of squares formula ๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐), we factor
๐๐ ๐๐
0.9(4๐ฅ๐ฅ4 โ ๐ฅ๐ฅ6) = 0.9[(2๐ฅ๐ฅ2)2 โ (๐ฅ๐ฅ3)2] = ๐๐.๐๐๏ฟฝ๐๐๐๐๐๐ + ๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐๏ฟฝ
c. Similarly as in the previous two examples, ๐ฅ๐ฅ4 โ 81 can be factored by following the difference of squares pattern. So,
๐ฅ๐ฅ4 โ 81 = (๐ฅ๐ฅ2)2 โ (9)2 = (๐ฅ๐ฅ2 + 9)(๐ฅ๐ฅ2 โ 9)
However, this factorization is not complete yet. Notice that ๐ฅ๐ฅ2 โ 9 is also a difference of squares, so the original polynomial can be factored further. Thus,
๐ฅ๐ฅ4 โ 81 = (๐ฅ๐ฅ2 + 9)(๐ฅ๐ฅ2 โ 9) = ๏ฟฝ๐๐๐๐ + ๐๐๏ฟฝ(๐๐ + ๐๐)(๐๐ โ ๐๐) Attention: The sum of squares, ๐ฅ๐ฅ2 + 9, cannot be factored using real coefficients.
Generally, except for a common factor, a quadratic binomial of the form ๐๐๐๐ + ๐๐๐๐ is not factorable over the real numbers.
d. Following the difference of squares formula, we have
16 โ (๐๐ โ 2)2 = 42 โ (๐๐ โ 2)2
= [4 + (๐๐ โ 2)][4 โ (๐๐ โ 2)]
= (4 + ๐๐ โ 2)(4 โ ๐๐ + 2) work out the inner brackets
= (๐๐ + ๐๐)(๐๐ โ ๐๐) combine like terms
Perfect Squares
Another frequently used special factoring formula is the perfect square of a sum or a difference.
๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐)๐๐ or
๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ = (๐๐ โ ๐๐)๐๐
Figure 3.2 shows the geometric interpretation of the perfect square of a sum. We encourage the reader to come up with a similar interpretation of the perfect square of a difference.
Remember to use brackets after the
negative sign!
Figure 3.2
๐๐ + ๐๐
๐๐ + ๐๐ ๐๐2 ๐๐๐๐
๐๐2 ๐๐๐๐
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section F2 | 235
Special Factoring and a General Strategy of Factoring
To factor a perfect square trinomial ๐๐๐๐ ยฑ ๐๐๐๐๐๐+ ๐๐๐๐, we find ๐๐ and ๐๐, which are the expressions being squared. Then, depending on the middle sign, we use ๐๐ and ๐๐ to form the perfect square of the sum (๐๐ + ๐๐)๐๐, or the perfect square of the difference (๐๐ โ ๐๐)๐๐.
Identifying Perfect Square Trinomials Decide whether the given polynomial is a perfect square.
a. 9๐ฅ๐ฅ2 + 6๐ฅ๐ฅ + 4 b. 9๐ฅ๐ฅ2 + 4๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ๐ฅ๐ฅ c. 25๐๐4 + 40๐๐2 โ 16 d. 49๐ฅ๐ฅ6 + 84๐ฅ๐ฅ๐ฅ๐ฅ3 + 36๐ฅ๐ฅ2 a. Observe that the outside terms of the trinomial 9๐ฅ๐ฅ2 + 6๐ฅ๐ฅ + 4 are perfect squares, as
9๐ฅ๐ฅ2 = (3๐ฅ๐ฅ)2 and 4 = 22. So, the trinomial would be a perfect square if the middle terms would equal 2 โ 3๐ฅ๐ฅ โ 2 = 12๐ฅ๐ฅ. Since this is not the case, our trinomial is not a perfect square.
Attention: Except for a common factor, trinomials of the type ๐๐๐๐ ยฑ ๐๐๐๐ + ๐๐๐๐ are not factorable over the real numbers!
b. First, we arrange the trinomial in decreasing order of the powers of ๐ฅ๐ฅ. So, we obtain
9๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ๐ฅ๐ฅ + 4๐ฅ๐ฅ2. Then, since 9๐ฅ๐ฅ2 = (3๐ฅ๐ฅ)2, 4๐ฅ๐ฅ2 = (2๐ฅ๐ฅ)2, and the middle term (except for the sign) equals 2 โ 3๐ฅ๐ฅ โ 2 = 12๐ฅ๐ฅ, we claim that the trinomial is a perfect square. Since the middle term is negative, this is the perfect square of a difference. So, the trinomial 9๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ๐ฅ๐ฅ + 4๐ฅ๐ฅ2 can be seen as
๐๐2 โ 2 ๐๐ ๐๐ + ๐๐2 = ( ๐๐ โ ๐๐ )2
(3๐ฅ๐ฅ)2 โ 2 โ 3๐ฅ๐ฅ โ 2๐ฅ๐ฅ + (2๐ฅ๐ฅ)2 = (3๐ฅ๐ฅ โ 2๐ฅ๐ฅ)2 c. Even though the coefficients of the trinomial 25๐๐4 + 40๐๐2 โ 16 and the distribution
of powers seem to follow the pattern of a perfect square, the last term is negative, which makes it not a perfect square.
d. Since 49๐ฅ๐ฅ6 = (7๐ฅ๐ฅ3)2, 36๐ฅ๐ฅ2 = (6๐ฅ๐ฅ)2, and the middle term equals 2 โ 7๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ =
84๐ฅ๐ฅ๐ฅ๐ฅ3, we claim that the trinomial is a perfect square. Since the middle term is positive, this is the perfect square of a sum. So, the trinomial 9๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ๐ฅ๐ฅ + 4๐ฅ๐ฅ2 can be seen as
๐๐2 + 2 ๐๐ ๐๐ + ๐๐2 = ( ๐๐ โ ๐๐ )2
(7๐ฅ๐ฅ3)2 + 2 โ 7๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ + (6๐ฅ๐ฅ)2 = (7๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ)2
Factoring Perfect Square Trinomials Factor each polynomial completely.
a. 25๐ฅ๐ฅ2 + 10๐ฅ๐ฅ + 1 b. ๐๐2 โ 12๐๐๐๐ + 36๐๐2 c. ๐๐2 โ 8๐๐ + 16 โ 49๐๐2 d. โ4๐ฅ๐ฅ2 โ 144๐ฅ๐ฅ8 + 48๐ฅ๐ฅ5
Solution
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section F3 | 236
Factoring
a. The outside terms of the trinomial 25๐ฅ๐ฅ2 + 10๐ฅ๐ฅ + 1 are perfect squares of 5๐๐ and 1, and the middle term equals 2 โ 5๐ฅ๐ฅ โ 1 = 10๐ฅ๐ฅ, so we can follow the perfect square formula. Therefore,
25๐ฅ๐ฅ2 + 10๐ฅ๐ฅ + 1 = (๐๐๐๐ + ๐๐)๐๐ b. The outside terms of the trinomial ๐๐2 โ 12๐๐๐๐ + 36๐๐2 are perfect squares of ๐๐ and 6๐๐,
and the middle term (disregarding the sign) equals 2 โ ๐๐ โ 6๐๐ = 12๐๐๐๐, so we can follow the perfect square formula. Therefore,
๐๐2 โ 12๐๐๐๐ + 36๐๐2 = (๐๐ โ ๐๐๐๐)๐๐ c. Observe that the first three terms of the polynomial ๐๐2 โ 8๐๐ + 16 โ 49๐๐2 form a
perfect square of ๐๐โ 6 and the last term is a perfect square of 7๐๐. So, we can write
๐๐2 โ 8๐๐ + 16 โ 49๐๐2 = (๐๐ โ 6)2 โ (7๐๐)2
Notice that this way we have formed a difference of squares. So we can factor it by following the difference of squares formula
(๐๐ โ 6)2 โ (7๐๐)2 = (๐๐โ ๐๐โ ๐๐๐ฉ๐ฉ)(๐๐โ ๐๐ + ๐๐๐ฉ๐ฉ)
d. As in any factoring problem, first we check the polynomial โ4๐ฅ๐ฅ2 โ 144๐ฅ๐ฅ8 + 48๐ฅ๐ฅ5 for a common factor, which is 4๐ฅ๐ฅ2. To leave the leading term of this polynomial positive, we factor out โ4๐ฅ๐ฅ2. So, we obtain
โ4๐ฅ๐ฅ2 โ 144๐ฅ๐ฅ8 + 48๐ฅ๐ฅ5
= โ4๐ฅ๐ฅ2 (1 + 36๐ฅ๐ฅ6 โ 12๐ฅ๐ฅ3)
= โ4๐ฅ๐ฅ2 (36๐ฅ๐ฅ6 โ 12๐ฅ๐ฅ3 + 1)
= โ๐๐๐๐๐๐ ๏ฟฝ๐๐๐๐๐๐ โ ๐๐๏ฟฝ๐๐
Sum or Difference of Cubes
The last special factoring formula to discuss in this section is the sum or difference of cubes.
๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐)๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐๏ฟฝ or
๐๐๐๐ โ ๐๐๐๐ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐๏ฟฝ
The reader is encouraged to confirm these formulas by multiplying the factors in the right-hand side of each equation. In addition, we offer a geometric visualization of one of these formulas, the difference of cubes, as shown in Figure 3.3.
Solution
fold to the perfect square form
arrange the polynomial in decreasing powers
This is not in factored form yet!
Figure 3.3
๐๐3 โ ๐๐3 = ๐๐2(๐๐ โ ๐๐)+๐๐๐๐(๐๐ โ ๐๐) + ๐๐2(๐๐ โ ๐๐)
๐๐
๐๐ โ ๐๐ ๐๐
๐๐
๐๐
๐๐ โ ๐๐
๐๐ โ ๐๐
๐๐
๐๐
๐๐3
= (๐๐ โ ๐๐)(๐๐2+๐๐๐๐ + ๐๐2)
๐๐3
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section F2 | 237
Special Factoring and a General Strategy of Factoring
Hints for memorization of the sum or difference of cubes formulas: โข The binomial factor is a copy of the sum or difference of the terms that were originally
cubed. โข The trinomial factor follows the pattern of a perfect square, except that the middle term
is single, not doubled. โข The signs in the factored form follow the pattern Same-Opposite-Positive (SOP).
Factoring Sums or Differences of Cubes Factor each polynomial completely.
a. 8๐ฅ๐ฅ3 + 1 b. 27๐ฅ๐ฅ7๐ฅ๐ฅ โ 125๐ฅ๐ฅ๐ฅ๐ฅ4 c. 2๐๐6 โ 128 d. (๐๐ โ 2)3 + ๐๐3
a. First, we rewrite each term of 8๐ฅ๐ฅ3 + 1 as a perfect cube of an expression.
๐๐ ๐๐
8๐ฅ๐ฅ3 + 1 = (2๐ฅ๐ฅ)2 + 12
Then, treating 2๐ฅ๐ฅ as the ๐๐ and 1 as the ๐๐ in the sum of cubes formula ๐๐3 + ๐๐3 =(๐๐ + ๐๐)(๐๐2 โ ๐๐๐๐ + ๐๐2), we factor:
๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐) ๏ฟฝ ๐๐๐๐ โ ๐๐ ๐๐ + ๐๐๐๐๏ฟฝ
8๐ฅ๐ฅ3 + 1 = (2๐ฅ๐ฅ)2 + 12 = (2๐ฅ๐ฅ + 1)((2๐ฅ๐ฅ)2 โ 2๐ฅ๐ฅ โ 1 + 12)
= (๐๐๐๐ + ๐๐)๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐ + ๐๐๏ฟฝ
Notice that the trinomial 4๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ + 1 in not factorable anymore. b. Since the two terms of the polynomial 27๐ฅ๐ฅ7๐ฅ๐ฅ โ 125๐ฅ๐ฅ๐ฅ๐ฅ4 contain the common factor
๐ฅ๐ฅ๐ฅ๐ฅ, we factor it out and obtain
27๐ฅ๐ฅ7๐ฅ๐ฅ โ 125๐ฅ๐ฅ๐ฅ๐ฅ4 = ๐ฅ๐ฅ๐ฅ๐ฅ(27๐ฅ๐ฅ6 โ 125๐ฅ๐ฅ3)
Observe that the remaining polynomial is a difference of cubes, (3๐ฅ๐ฅ2)3 โ (5๐ฅ๐ฅ)3. So, we factor,
27๐ฅ๐ฅ7๐ฅ๐ฅ โ 125๐ฅ๐ฅ๐ฅ๐ฅ4 = ๐ฅ๐ฅ๐ฅ๐ฅ[(3๐ฅ๐ฅ2)3 โ (5๐ฅ๐ฅ)3]
( ๐๐ + ๐๐ ) ๏ฟฝ ๐๐๐๐ โ ๐๐ ๐๐ + ๐๐๐๐๏ฟฝ
= ๐ฅ๐ฅ๐ฅ๐ฅ(3๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ)[(3๐ฅ๐ฅ2)2 + 3๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ + (5๐ฅ๐ฅ)2]
= ๐๐๐๐๏ฟฝ๐๐๐๐๐๐ โ ๐๐๐๐๏ฟฝ๏ฟฝ๐๐๐๐๐๐ + ๐๐๐๐๐๐๐๐๐๐+ ๐๐๐๐๐๐๐๐๏ฟฝ c. After factoring out the common factor 2, we obtain
2๐๐6 โ 128 = 2(๐๐6 โ 64)
Solution
Difference of squares or difference of cubes?
Quadratic trinomials of the form ๐๐๐๐ ยฑ ๐๐๐๐ + ๐๐๐๐
are not factorable!
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section F3 | 238
Factoring
Notice that ๐๐6 โ 64 can be seen either as a difference of squares, (๐๐3)2 โ 82, or as a difference of cubes, (๐๐2)3 โ 43. It turns out that applying the difference of squares formula first leads us to a complete factorization while starting with the difference of cubes does not work so well here. See the two approaches below.
(๐๐3)2 โ 82 (๐๐2)3 โ 43
= (๐๐3 + 8)(๐๐3 โ 8) = (๐๐2 โ 4)(๐๐4 + 4๐๐2 + 16) = (๐๐ + 2)(๐๐2 โ 2๐๐ + 4)(๐๐ โ 2)(๐๐2 + 2๐๐ + 4) = (๐๐ + 2)(๐๐ โ 2)(๐๐4 + 4๐๐2 + 16)
Therefore, the original polynomial should be factored as follows:
2๐๐6 โ 128 = 2(๐๐6 โ 64) = 2[(๐๐3)2 โ 82] = 2(๐๐3 + 8)(๐๐3 โ 8)
= ๐๐(๐ฉ๐ฉ + ๐๐)๏ฟฝ๐ฉ๐ฉ๐๐ โ ๐๐๐ฉ๐ฉ + ๐๐๏ฟฝ(๐ฉ๐ฉ โ ๐๐)๏ฟฝ๐ฉ๐ฉ๐๐ + ๐๐๐ฉ๐ฉ + ๐๐๏ฟฝ
d. To factor (๐๐ โ 2)3 + ๐๐3, we follow the sum of cubes formula (๐๐ + ๐๐)(๐๐2 โ ๐๐๐๐ + ๐๐2) by assuming ๐๐ = ๐๐ โ 2 and ๐๐ = ๐๐. So, we have
(๐๐ โ 2)3 + ๐๐3 = (๐๐ โ 2 + ๐๐) [(๐๐ โ 2)2 โ (๐๐ โ 2)๐๐ + ๐๐2]
= (๐๐ โ 2 + ๐๐) [๐๐2 โ 2๐๐๐๐ + 4 โ ๐๐๐๐ + 2๐๐ + ๐๐2]
= (๐๐ โ ๐๐ + ๐๐) ๏ฟฝ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐ + ๐๐๐๐ + ๐๐๐๐๏ฟฝ
General Strategy of Factoring
Recall that a polynomial with integral coefficients is factored completely if all of its factors are prime over the integers.
How to Factorize Polynomials Completely?
1. Factor out all common factors. Leave the remaining polynomial with a positive leading term and integral coefficients, if possible.
2. Check the number of terms. If the polynomial has
- more than three terms, try to factor by grouping; a four term polynomial may require 2-2, 3-1, or 1-3 types of grouping.
- three terms, factor by guessing, decomposition, or follow the perfect square formula, if applicable.
- two terms, follow the difference of squares, or sum or difference of cubes formula, if applicable. Remember that sum of squares, ๐๐๐๐ + ๐๐๐๐, is not factorable over the real numbers, except for possibly a common factor.
There is no easy way of factoring this trinomial!
4 prime factors, so the factorization is complete
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section F2 | 239
Special Factoring and a General Strategy of Factoring
3. Keep in mind the special factoring formulas:
Difference of Squares ๐๐๐๐ โ ๐๐๐๐ = (๐๐ + ๐๐)(๐๐ โ ๐๐) Perfect Square of a Sum ๐๐๐๐ + ๐๐๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐)๐๐ Perfect Square of a Difference ๐๐๐๐ โ ๐๐๐๐๐๐ + ๐๐๐๐ = (๐๐ โ ๐๐)๐๐ Sum of Cubes ๐๐๐๐ + ๐๐๐๐ = (๐๐ + ๐๐)๏ฟฝ๐๐๐๐ โ ๐๐๐๐ + ๐๐๐๐๏ฟฝ Difference of Cubes ๐๐๐๐ โ ๐๐๐๐ = (๐๐ โ ๐๐)๏ฟฝ๐๐๐๐ + ๐๐๐๐ + ๐๐๐๐๏ฟฝ 4. Keep factoring each of the obtained factors until all of them are prime over the
integers.
Multiple-step Factorization Factor each polynomial completely.
a. 80๐ฅ๐ฅ5 โ 5๐ฅ๐ฅ b. 4๐๐2 โ 4๐๐ + 1 โ ๐๐2 c. (5๐๐ + 8)2 โ 6(5๐๐ + 8) + 9 d. (๐๐ โ 2๐๐)3 + (๐๐ + 2๐๐)3
a. First, we factor out the GCF of 80๐ฅ๐ฅ5 and โ5๐ฅ๐ฅ, which equals to 5๐ฅ๐ฅ. So, we obtain
80๐ฅ๐ฅ5 โ 5๐ฅ๐ฅ = 5๐ฅ๐ฅ(16๐ฅ๐ฅ4 โ 1)
Then, we notice that 16๐ฅ๐ฅ4 โ 1 can be seen as the difference of squares (4๐ฅ๐ฅ2)2 โ 12. So, we factor further
80๐ฅ๐ฅ5 โ 5๐ฅ๐ฅ = 5๐ฅ๐ฅ(4๐ฅ๐ฅ2 + 1)(4๐ฅ๐ฅ2 โ 1)
The first binomial factor, 4๐ฅ๐ฅ2 + 1, cannot be factored any further using integral coefficients as it is the sum of squares, (2๐ฅ๐ฅ)2 + 12. However, the second binomial factor, 4๐ฅ๐ฅ2 โ 1, is still factorable as a difference of squares, (2๐ฅ๐ฅ)2 โ 12. Therefore,
80๐ฅ๐ฅ5 โ 5๐ฅ๐ฅ = ๐๐๐๐๏ฟฝ๐๐๐๐๐๐ + ๐๐๏ฟฝ(๐๐๐๐ + ๐๐)(๐๐๐๐ โ ๐๐)
This is a complete factorization as all the factors are prime over the integers. b. The polynomial 4๐๐2 โ 4๐๐ + 1 โ ๐๐2 consists of four terms, so we might be able to
factor it by grouping. Observe that the 2-2 type of grouping has no chance to succeed, as the first two terms involve only the variable ๐๐ while the second two terms involve only the variable ๐๐. This means that after factoring out the common factor in each group, the remaining binomials would not be the same. So, the 2-2 grouping would not lead us to a factorization. However, the 3-1 type of grouping should help. This is because the first three terms form the perfect square, (2๐๐ โ 1)2, and there is a subtraction before the last term ๐๐2, which is also a perfect square. So, in the end, we can follow the difference of squares formula to complete the factoring process.
4๐๐2 โ 4๐๐ + 1๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝ๏ฟฝโ ๐๐2๏ฟฝ = (2๐๐ โ 1)2 โ ๐๐2 = (๐๐๐๐ โ ๐๐ โ ๐๐)(๐๐๐๐ โ ๐๐ + ๐๐)
Solution
3-1 type of grouping
repeated difference of squares
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section F3 | 240
Factoring
c. To factor (5๐๐ + 8)2 โ 6(5๐๐ + 8) + 9, it is convenient to substitute a new variable, say ๐๐, for the expression 5๐๐ + 8. Then,
(5๐๐ + 8)2 โ 6(5๐๐ + 8) + 9 = ๐๐2 โ 6๐๐ + 9
= (๐๐ โ 3)2
= (5๐๐ + 8 โ 3)2
= (5๐๐ + 5)2
Notice that 5๐๐ + 5 can still be factored by taking the 5 out. So, for a complete factorization, we factor further
(5๐๐ + 5)2 = ๏ฟฝ5(๐๐ + 1)๏ฟฝ2 = ๐๐๐๐(๐๐ + ๐๐)๐๐
d. To factor (๐๐ โ 2๐๐)3 + (๐๐ + 2๐๐)3, we follow the sum of cubes formula (๐๐ + ๐๐)(๐๐2 โ
๐๐๐๐ + ๐๐2) by assuming ๐๐ = ๐๐ โ 2๐๐ and ๐๐ = ๐๐ + 2๐๐. So, we have
(๐๐ โ 2๐๐)3 + (๐๐ + 2๐๐)3
= (๐๐ โ 2๐๐ + ๐๐ + 2๐๐) [(๐๐ โ 2๐๐)2 โ (๐๐ โ 2๐๐)(๐๐ + 2๐๐) + (๐๐ + 2๐๐)2]
= 2๐๐ [๐๐2 โ 4๐๐๐๐ + 4๐๐2 โ (๐๐2 โ 4๐๐2) + ๐๐2 + 4๐๐๐๐ + 4๐๐2]
= 2๐๐ (2๐๐2 + 8๐๐2 โ ๐๐2 + 4๐๐2) = ๐๐๐๐๏ฟฝ๐๐๐๐ + ๐๐๐๐๐๐๐๐๏ฟฝ
F.3 Exercises
Vocabulary Check Complete each blank with one of the suggested words, or the most appropriate term or
phrase from the given list: difference of cubes, difference of squares, perfect square, sum of cubes, sum of squares.
1. If a binomial is a ________________________________ its factorization has the form (๐๐ + ๐๐)(๐๐ โ ๐๐).
2. Trinomials of the form ๐๐2 ยฑ 2๐๐๐๐ + ๐๐2 are ____________________________ trinomials.
3. The product (๐๐ + ๐๐)(๐๐2 โ ๐๐๐๐ + ๐๐2) is the factorization of the __________________________.
4. The product (๐๐ โ ๐๐)(๐๐2 + ๐๐๐๐ + ๐๐2) is the factorization of the __________________________.
5. A ________________________ is not factorable.
6. Quadratic trinomials of the form ๐๐2 ยฑ ๐๐๐๐ + ๐๐2 _________________๐๐๐๐๐๐ /๐๐๐๐๐๐ ๐๐๐๐๐ก๐ก
factorable.
go back to the original variable
perfect square! factoring by substitution
multiple special formulas and simplifying
Remember to represent the new variable by a
different letter than the original variable!
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section F2 | 241
Special Factoring and a General Strategy of Factoring
Concept Check Determine whether each polynomial is a perfect square, a difference of squares, a sum or difference of cubes, or neither.
7. 0.25๐ฅ๐ฅ2 โ 0.16๐ฅ๐ฅ2 8. ๐ฅ๐ฅ2 โ 14๐ฅ๐ฅ + 49
9. 9๐ฅ๐ฅ4 + 4๐ฅ๐ฅ2 + 1 10. 4๐ฅ๐ฅ2 โ (๐ฅ๐ฅ + 4)2
11. 125๐ฅ๐ฅ3 โ 64 12. ๐ฅ๐ฅ12 + 0.008๐ฅ๐ฅ3
13. โ๐ฅ๐ฅ4 + 16๐ฅ๐ฅ4 14. 64 + 48๐ฅ๐ฅ3 + 9๐ฅ๐ฅ6
15. 25๐ฅ๐ฅ6 โ 10๐ฅ๐ฅ3๐ฅ๐ฅ2 + ๐ฅ๐ฅ4 16. โ4๐ฅ๐ฅ6 โ ๐ฅ๐ฅ6
17. โ8๐ฅ๐ฅ3 + 27๐ฅ๐ฅ6 18. 81๐ฅ๐ฅ2 โ 16๐ฅ๐ฅ
Concept Check
19. The binomial 4๐ฅ๐ฅ2 + 64 is an example of a sum of two squares that can be factored. Under what conditions can the sum of two squares be factored?
20. Insert the correct signs into the blanks. a. 8 + ๐๐3 = (2 ___ ๐๐)(4 ___ 2๐๐ ___ ๐๐2) b. ๐๐3 โ 1 = (๐๐ ___ 1)(๐๐2 ___ ๐๐ ___ 1) Factor each polynomial completely, if possible.
21. ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ2 22. ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ๐ฅ๐ฅ + ๐ฅ๐ฅ2 23. ๐ฅ๐ฅ3 โ ๐ฅ๐ฅ3
24. 16๐ฅ๐ฅ2 โ 100 25. 4๐ง๐ง2 โ 4๐ง๐ง + 1 26. ๐ฅ๐ฅ3 + 27
27. 4๐ง๐ง2 + 25 28. ๐ฅ๐ฅ2 + 18๐ฅ๐ฅ + 81 29. 125โ ๐ฅ๐ฅ3
30. 144๐ฅ๐ฅ2 โ 64๐ฅ๐ฅ2 31. ๐๐2 + 20๐๐๐๐ + 100๐๐2 32. 27๐๐3๐๐6 + 1
33. 9๐๐4 โ 25๐๐6 34. 25 โ 40๐ฅ๐ฅ + 16๐ฅ๐ฅ2 35. ๐๐6 โ 64๐๐3
36. 16๐ฅ๐ฅ2๐ง๐ง2 โ 100๐ฅ๐ฅ2 37. 4 + 49๐๐2 + 28๐๐ 38. ๐ฅ๐ฅ12 + 0.008๐ฅ๐ฅ3
39. ๐๐4 โ 9๐๐2 40. 9๐๐2 โ 12๐๐๐๐ โ 4๐๐2 41. 18โ ๐๐3
42. 0.04๐ฅ๐ฅ2 โ 0.09๐ฅ๐ฅ2 43. ๐ฅ๐ฅ4 + 8๐ฅ๐ฅ2 + 1 44. โ 127
+ ๐ก๐ก3
45. 16๐ฅ๐ฅ6 โ 121๐ฅ๐ฅ2๐ฅ๐ฅ4 46. 9 + 60๐๐๐๐ + 100๐๐2๐๐2 47. โ๐๐3๐๐3 โ 125๐๐6
48. 36๐๐2๐๐ โ 1 49. 9๐๐8 โ 48๐๐4๐๐ + 64๐๐2 50. 9๐ฅ๐ฅ3 + 8
51. (๐ฅ๐ฅ + 1)2 โ 49 52. 14๐ข๐ข2 โ ๐ข๐ข๐ข๐ข + ๐ข๐ข2 53. 2๐ก๐ก4 โ 128๐ก๐ก
54. 81 โ (๐๐ + 3)2 55. ๐ฅ๐ฅ2๐๐ + 6๐ฅ๐ฅ๐๐ + 9 56. 8 โ (๐๐ + 2)3
57. 16๐ง๐ง4 โ 1 58. 5๐๐3 + 20๐๐2 + 20๐๐ 59. (๐ฅ๐ฅ + 5)3 โ ๐ฅ๐ฅ3
60. ๐๐4 โ 81๐๐4 61. 0.25๐ง๐ง2 โ 0.7๐ง๐ง + 0.49 62. (๐ฅ๐ฅ โ 1)3 + (๐ฅ๐ฅ + 1)3
63. (๐ฅ๐ฅ โ 2๐ฅ๐ฅ)2 โ (๐ฅ๐ฅ + ๐ฅ๐ฅ)2 64. 0.81๐๐8 + 9๐๐4 + 25 65. (๐ฅ๐ฅ + 2)3 โ (๐ฅ๐ฅ โ 2)3
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section F3 | 242
Factoring
Factor each polynomial completely.
66. 3๐ฅ๐ฅ3 โ 12๐ฅ๐ฅ2๐ฅ๐ฅ 67. 2๐ฅ๐ฅ2 + 50๐๐2 โ 20๐๐๐ฅ๐ฅ 68. ๐ฅ๐ฅ3 โ ๐ฅ๐ฅ๐ฅ๐ฅ2 + ๐ฅ๐ฅ2๐ฅ๐ฅ โ ๐ฅ๐ฅ3
69. ๐ฅ๐ฅ2 โ 9๐๐2 + 12๐ฅ๐ฅ + 36 70. 64๐ข๐ข6 โ 1 71. 7๐๐3 + ๐๐6 โ 8
72. โ7๐๐2 + 2๐๐3 + 4๐๐ โ 14 73. ๐๐8 โ ๐๐8 74. ๐ฅ๐ฅ9 โ ๐ฅ๐ฅ
75. (๐ฅ๐ฅ2 โ 2)2 โ 4(๐ฅ๐ฅ2 โ 2) โ 21 76. 8(๐๐ โ 3)2 โ 64(๐๐ โ 3) + 128 77. ๐๐2 โ ๐๐2 โ 6๐๐ โ 9
78. 25(2๐๐ โ ๐๐)2 โ 9 79. 3๐ฅ๐ฅ2๐ฅ๐ฅ2๐ง๐ง + 25๐ฅ๐ฅ๐ฅ๐ฅ๐ง๐ง2 + 28๐ง๐ง3 80. ๐ฅ๐ฅ8๐๐ โ ๐ฅ๐ฅ2
81. ๐ฅ๐ฅ6 โ 2๐ฅ๐ฅ5 + ๐ฅ๐ฅ4 โ ๐ฅ๐ฅ2 + 2๐ฅ๐ฅ โ 1 82. 4๐ฅ๐ฅ2๐ฅ๐ฅ4 โ 9๐ฅ๐ฅ4 โ 4๐ฅ๐ฅ2๐ง๐ง4 + 9๐ง๐ง4 83. ๐๐2๐ค๐ค+1 + 2๐๐๐ค๐ค+1 + ๐๐
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section F4 | 243
Solving Polynomial Equations and Applications of Factoring
F.4 Solving Polynomial Equations and Applications of Factoring
Many application problems involve solving polynomial equations. In Chapter L, we studied methods for solving linear, or first-degree, equations. Solving higher degree polynomial equations requires other methods, which often involve factoring. In this chapter, we study solving polynomial equations using the zero-product property, graphical connections between roots of an equation and zeros of the corresponding function, and some application problems involving polynomial equations or formulas that can be solved by factoring.
Zero-Product Property
Recall that to solve a linear equation, for example 2๐ฅ๐ฅ + 1 = 0, it is enough to isolate the variable on one side of the equation by applying reverse operations. Unfortunately, this method usually does not work when solving higher degree polynomial equations. For example, we would not be able to solve the equation ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ = 0 through the reverse operation process, because the variable ๐ฅ๐ฅ appears in different powers.
So โฆ how else can we solve it?
In this particular example, it is possible to guess the solutions. They are ๐ฅ๐ฅ = 0 and ๐ฅ๐ฅ = 1.
But how can we solve it algebraically?
It turns out that factoring the left-hand side of the equation ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ = 0 helps. Indeed, ๐ฅ๐ฅ(๐ฅ๐ฅ โ 1) = 0 tells us that the product of ๐ฅ๐ฅ and ๐ฅ๐ฅ โ 1 is 0. Since the product of two quantities is 0, at least one of them must be 0. So, either ๐ฅ๐ฅ = 0 or ๐ฅ๐ฅ โ 1 = 0, which solves to ๐ฅ๐ฅ = 1.
The equation discussed above is an example of a second degree polynomial equation, more commonly known as a quadratic equation.
Definition 4.1 A quadratic equation is a second degree polynomial equation in one variable that can be written in the form,
๐๐๐๐๐๐ + ๐๐๐๐ + ๐๐ = ๐๐,
where ๐๐, ๐๐, and ๐๐ are real numbers and ๐๐ โ 0. This form is called standard form.
One of the methods of solving such equations involves factoring and the zero-product property that is stated below.
For any real numbers ๐๐ and ๐๐,
๐๐๐๐ = ๐๐ if and only if ๐๐ = ๐๐ or ๐๐ = ๐๐
This means that any product containing a factor of 0 is equal to 0, and conversely, if a product is equal to 0, then at least one of its factors is equal to 0.
The implication โif ๐๐ = ๐๐ or ๐๐ = ๐๐, then ๐๐๐๐ = ๐๐โ is true by the multiplicative property of zero.
To prove the implication โif ๐๐๐๐ = ๐๐, then ๐๐ = ๐๐ or ๐๐ = ๐๐โ, let us assume first that ๐๐ โ 0. (As, if ๐๐ = 0, then the implication is already proven.)
Zero-Product Theorem
Proof
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section F4 | 244
Factoring
Since ๐๐ โ 0, then 1๐๐ exists. Therefore, both sides of ๐๐๐๐ = 0 can be multiplied by 1
๐๐ and we
obtain
1๐๐โ ๐๐๐๐ =
1๐๐โ 0
๐๐ = 0,
which concludes the proof.
Attention: The zero-product property works only for a product equal to 0. For example, the fact that ๐๐๐๐ = ๐๐ does not mean that either ๐๐ or ๐๐ equals to 1.
Using the Zero-Product Property to Solve Polynomial Equations
Solve each equation.
a. (๐ฅ๐ฅ โ 3)(2๐ฅ๐ฅ + 5) = 0 b. 2๐ฅ๐ฅ(๐ฅ๐ฅ โ 5)2 = 0 a. Since the product of ๐ฅ๐ฅ โ 3 and 2๐ฅ๐ฅ + 5 is equal to zero, then by the zero-product
property at least one of these expressions must equal to zero. So,
๐ฅ๐ฅ โ 3 = 0 or 2๐ฅ๐ฅ + 5 = 0
which results in ๐ฅ๐ฅ = 3 or 2๐ฅ๐ฅ = โ5 ๐ฅ๐ฅ = โ5
2
Thus, ๏ฟฝโ ๐๐๐๐
,๐๐๏ฟฝ is the solution set of the given equation.
b. Since the product 2๐ฅ๐ฅ(๐ฅ๐ฅ โ 5)2 is zero, then either ๐ฅ๐ฅ = 0 or ๐ฅ๐ฅ โ 5 = 0, which solves to ๐ฅ๐ฅ = 5. Thus, the solution set is equal to {๐๐,๐๐}.
Note 1: The factor of 2 does not produce any solution, as 2 is never equal to 0.
Note 2: The perfect square (๐ฅ๐ฅ โ 5)2 equals to 0 if and only if the base ๐ฅ๐ฅ โ 5 equals to 0.
Solving Polynomial Equations by Factoring
To solve polynomial equations of second or higher degree by factoring, we
โข arrange the polynomial in decreasing order of powers on one side of the equation, โข keep the other side of the equation equal to 0, โข factor the polynomial completely, โข use the zero-product property to form linear equations for each factor, โข solve the linear equations to find the roots (solutions) to the original equation.
Solution
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section F4 | 245
Solving Polynomial Equations and Applications of Factoring
Solving Quadratic Equations by Factoring Solve each equation by factoring.
a. ๐ฅ๐ฅ2 + 9 = 6๐ฅ๐ฅ b. 15๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ = 0
c. (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 1) = 4(3 โ ๐ฅ๐ฅ) โ 8 d. (๐ฅ๐ฅ โ 3)2 = 36๐ฅ๐ฅ2
a. To solve ๐ฅ๐ฅ2 + 9 = 6๐ฅ๐ฅ by factoring we need one side of this equation equal to 0. So, first, we move the 6๐ฅ๐ฅ term to the left side of the equation,
๐ฅ๐ฅ2 + 9 โ 6๐ฅ๐ฅ = 0,
and arrange the terms in decreasing order of powers of ๐ฅ๐ฅ,
๐ฅ๐ฅ2 โ 6๐ฅ๐ฅ + 9 = 0.
Then, by observing that the resulting trinomial forms a perfect square of ๐ฅ๐ฅ โ 3, we factor
(๐ฅ๐ฅ โ 3)2 = 0, which is equivalent to
๐ฅ๐ฅ โ 3 = 0, and finally
๐ฅ๐ฅ = 3.
So, the solution is ๐ฅ๐ฅ = ๐๐.
b. After factoring the left side of the equation 15๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ = 0,
3๐ฅ๐ฅ(5๐ฅ๐ฅ โ 4) = 0,
we use the zero-product property. Since 3 is never zero, the solutions come from the equations
๐ฅ๐ฅ = ๐๐ or 5๐ฅ๐ฅ โ 4 = 0.
Solving the second equation for ๐ฅ๐ฅ, we obtain
5๐ฅ๐ฅ = 4, and finally
๐ฅ๐ฅ = ๐๐๐๐.
So, the solution set consists of 0 and ๐๐๐๐.
c. To solve (๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 1) = 4(3 โ ๐ฅ๐ฅ) โ 8 by factoring, first, we work out the brackets and arrange the polynomial in decreasing order of exponents on the left side of the equation. So, we obtain
๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 2 = 12โ 4๐ฅ๐ฅ โ 8
๐ฅ๐ฅ2 + 5๐ฅ๐ฅ โ 6 = 0
(๐ฅ๐ฅ + 6)(๐ฅ๐ฅ โ 1) = 0
Solution
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section F4 | 246
Factoring
Now, we can read the solutions from each bracket, that is, ๐ฅ๐ฅ = โ๐๐ and ๐ฅ๐ฅ = ๐๐.
Observation: In the process of solving a linear equation of the form ๐๐๐ฅ๐ฅ + ๐๐ = 0, first we subtract ๐๐ and then we divide by ๐๐. So the solution, sometimes
referred to as the root, is ๐ฅ๐ฅ = โ๐๐๐๐. This allows us to read the solution
directly from the equation. For example, the solution to ๐ฅ๐ฅ โ 1 = 0 is ๐ฅ๐ฅ = 1 and the solution to 2๐ฅ๐ฅ โ 1 = 0 is ๐ฅ๐ฅ = 1
2.
d. To solve (๐ฅ๐ฅ โ 3)2 = 36๐ฅ๐ฅ2, we bring all the terms to one side and factor the obtained difference of squares, following the formula ๐๐2 โ ๐๐2 = (๐๐ + ๐๐)(๐๐ โ ๐๐). So, we have
(๐ฅ๐ฅ โ 3)2 โ 36๐ฅ๐ฅ2 = 0
(๐ฅ๐ฅ โ 3 + 6๐ฅ๐ฅ)(๐ฅ๐ฅ โ 3 โ 6๐ฅ๐ฅ) = 0
(7๐ฅ๐ฅ โ 3)(โ5๐ฅ๐ฅ โ 3) = 0
Then, by the zero-product property,
7๐ฅ๐ฅ โ 3 = 0 or โ5๐ฅ๐ฅ โ 3, which results in
๐ฅ๐ฅ = ๐๐๐๐ or ๐ฅ๐ฅ = โ๐๐
๐๐.
Solving Polynomial Equations by Factoring
Solve each equation by factoring.
a. 2๐ฅ๐ฅ3 โ 2๐ฅ๐ฅ2 = 12๐ฅ๐ฅ b. ๐ฅ๐ฅ4 + 36 = 13๐ฅ๐ฅ2
a. First, we bring all the terms to one side of the equation and then factor the resulting polynomial.
2๐ฅ๐ฅ3 โ 2๐ฅ๐ฅ2 = 12๐ฅ๐ฅ
2๐ฅ๐ฅ3 โ 2๐ฅ๐ฅ2 โ 12๐ฅ๐ฅ = 0
2๐ฅ๐ฅ(๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 6) = 0
2๐ฅ๐ฅ(๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 2) = 0
By the zero-product property, the factors ๐ฅ๐ฅ, (๐ฅ๐ฅ โ 3) and (๐ฅ๐ฅ + 2), give us the corresponding solutions, 0, 3, and โ 2. So, the solution set of the given equation is {๐๐,๐๐,โ๐๐}.
b. Similarly as in the previous examples, we solve ๐ฅ๐ฅ4 + 36 = 13๐ฅ๐ฅ2 by factoring and
using the zero-product property. Since
๐ฅ๐ฅ4 โ 13๐ฅ๐ฅ2 + 36 = 0
Solution
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section F4 | 247
Solving Polynomial Equations and Applications of Factoring
(๐ฅ๐ฅ2 โ 4)(๐ฅ๐ฅ2 โ 9) = 0
(๐ฅ๐ฅ + 2)(๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 3)(๐ฅ๐ฅ โ 3) = 0,
then, the solution set of the original equation is {โ๐๐,๐๐,โ๐๐,๐๐}
Observation: ๐๐-th degree polynomial equations may have up to ๐๐ roots (solutions).
Factoring in Applied Problems
Factoring is a useful strategy when solving applied problems. For example, factoring is often used in solving formulas for a variable, in finding roots of a polynomial function, and generally, in any problem involving polynomial equations that can be solved by factoring.
Solving Formulas with the Use of Factoring
Solve each formula for the specified variable.
a. ๐ด๐ด = 2๐๐๐ค๐ค + 2๐ค๐ค๐ค๐ค + 2๐ค๐ค๐๐, for ๐๐ b. ๐ ๐ = 2๐๐+3๐๐
, for ๐๐ a. To solve ๐ด๐ด = 2๐๐๐ค๐ค + 2๐ค๐ค๐ค๐ค + 2๐ค๐ค๐๐ for ๐๐, we want to keep both terms containing ๐๐ on
the same side of the equation and bring the remaining terms to the other side. Here is an equivalent equation,
๐ด๐ด โ 2๐ค๐ค๐ค๐ค = 2๐๐๐ค๐ค + 2๐ค๐ค๐๐,
which, for convenience, could be written starting with โ-terms:
2๐๐๐ค๐ค + 2๐ค๐ค๐๐ = ๐ด๐ด โ 2๐ค๐ค๐ค๐ค
Now, factoring ๐๐ out causes that ๐๐ appears in only one place, which is what we need to isolate it. So,
(2๐ค๐ค + 2๐ค๐ค)๐๐ = ๐ด๐ด โ 2๐ค๐ค๐ค๐ค
๐๐ =๐จ๐จ โ ๐๐๐๐๐๐๐๐๐๐ + ๐๐๐๐
Notice: In the above formula, there is nothing that can be simplified. Trying to reduce 2 or 2๐ค๐ค or ๐ค๐ค would be an error, as there is no essential common factor that can be carried out of the numerator.
b. When solving ๐ ๐ = 2๐๐+3๐๐
for ๐๐, our goal is to, firstly, keep the variable ๐๐ in the numerator and secondly, to keep it in a single place. So, we have
๐ ๐ =2๐๐ + 3๐๐
๐ ๐ ๐๐ = 2๐๐ + 3
Solution
/ รท (2๐ค๐ค + 2๐ค๐ค)
/ โ ๐ก๐ก
/ โ2๐ก๐ก
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section F4 | 248
Factoring
๐ ๐ ๐๐ โ 2๐๐ = 3
๐๐(๐ ๐ โ 2) = 3
๐๐ =๐๐
๐๐ โ ๐๐.
Finding Roots of a Polynomial Function
A small rocket is launched from the ground vertically upwards with an initial velocity of 128 feet per second. Its height in feet after ๐ก๐ก seconds is a function defined by
โ(๐ก๐ก) = โ16๐ก๐ก2 + 128๐ก๐ก.
After how many seconds will the rocket hit the ground? The rocket hits the ground when its height is 0. So, we need to find the time ๐ก๐ก for which โ(๐ก๐ก) = 0. Therefore, we solve the equation
โ16๐ก๐ก2 + 128๐ก๐ก = 0 for ๐ก๐ก. From the factored form
โ16๐ก๐ก(๐ก๐ก โ 8) = 0
we conclude that the rocket is on the ground at times 0 and 8 seconds. So the rocket hits the ground after 8 seconds from its launch.
Solving an Application Problem with the Use of Factoring
The height of a triangle is 1 meter less than twice the length of the base. The area is 14 m2. What are the measures of the base and the height? Let ๐๐ and โ represent the base and the height of the triangle, correspondingly. The first sentence states that โ is 1 less than 2 times ๐๐. So, we record
โ = 2๐๐ โ 1.
Using the formula for area of a triangle, ๐ด๐ด = 12๐๐โ, and the fact that ๐ด๐ด = 14, we obtain
14 =12๐๐(2๐๐ โ 1).
Since this is a one-variable quadratic equation, we will attempt to solve it by factoring, after bringing all the terms to one side of the equation. So, we have
0 =12๐๐(2๐๐ โ 1) โ 14
0 = ๐๐(2๐๐ โ 1) โ 28
0 = 2๐๐2 โ ๐๐ โ 28
Solution
Solution
/ โ 2
to clear the fraction, multiply each term by 2 before working out the bracket
factor ๐ก๐ก
/ รท (๐ ๐ โ 2)
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section F4 | 249
Solving Polynomial Equations and Applications of Factoring
0 = (2๐๐ + 7)(๐๐ โ 4),
which by the zero-product property gives us ๐๐ = โ72 or ๐๐ = 4. Since ๐๐ represents the length
of the base, it must be positive. So, the base is 4 meters long and the height is โ = 2๐๐ โ1 = 2 โ 4 โ 1 = ๐๐ meters long.
F.4 Exercises
Vocabulary Check Complete each blank with the most appropriate term from the given list: factored, linear,
n, zero, zero-product.
1. The ________________________ property states that if ๐๐๐๐ = 0 then either ๐๐ = 0 or ๐๐ = 0.
2. When a quadratic equation is solved by factoring, the zero-product property is used to form two ______________ equations.
3. The zero-product property can be applied only when one side of the equation is equal to __________ and the other side is in a ________________ form.
4. An ๐๐-th degree polynomial equation may have up to _____ solutions. Concept Check True or false.
5. If ๐ฅ๐ฅ๐ฅ๐ฅ = 0 then ๐ฅ๐ฅ = 0 or ๐ฅ๐ฅ = 0.
6. If ๐๐๐๐ = 1 then ๐๐ = 1 or ๐๐ = 1.
7. If ๐ฅ๐ฅ + ๐ฅ๐ฅ = 0 then ๐ฅ๐ฅ = 0 or ๐ฅ๐ฅ = 0.
8. If ๐๐2 = 0 then ๐๐ = 0.
9. If ๐ฅ๐ฅ2 = 1 then ๐ฅ๐ฅ = 1. Concept Check
10. Which of the following equations is not in proper form for using the zero-product property.
a. ๐ฅ๐ฅ(๐ฅ๐ฅ โ 1) + 3(๐ฅ๐ฅ โ 1) = 0 b. (๐ฅ๐ฅ + 3)(๐ฅ๐ฅ โ 1) = 0
c. ๐ฅ๐ฅ(๐ฅ๐ฅ โ 1) = 3(๐ฅ๐ฅ โ 1) d. (๐ฅ๐ฅ + 3)(๐ฅ๐ฅ โ 1) = โ3 Solve each equation.
11. 3(๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ + 4) = 0 12. 2(๐ฅ๐ฅ + 5)(๐ฅ๐ฅ โ 7) = 0
13. (3๐ฅ๐ฅ + 1)(5๐ฅ๐ฅ + 4) = 0 14. (2๐ฅ๐ฅ โ 3)(4๐ฅ๐ฅ โ 1) = 0
15. ๐ฅ๐ฅ2 + 9๐ฅ๐ฅ + 18 = 0 16. ๐ฅ๐ฅ2 โ 18๐ฅ๐ฅ + 80 = 0
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section F4 | 250
Factoring
17. 2๐ฅ๐ฅ2 = 7 โ 5๐ฅ๐ฅ 18. 3๐๐2 = 14๐๐ โ 8
19. ๐ฅ๐ฅ2 + 6๐ฅ๐ฅ = 0 20. 6๐ฅ๐ฅ2 โ 3๐ฅ๐ฅ = 0
21. (4 โ ๐๐)2 = 0 22. (2๐๐ + 5)2 = 0
23. 0 = 4๐๐2 โ 20๐๐ + 25 24. 0 = 16๐ฅ๐ฅ2 + 8๐ฅ๐ฅ + 1
25. ๐๐2 โ 32 = โ4๐๐ 26. 19๐๐ + 36 = 6๐๐2
27. ๐ฅ๐ฅ2 + 3 = 10๐ฅ๐ฅ โ 2๐ฅ๐ฅ2 28. 3๐ฅ๐ฅ2 + 9๐ฅ๐ฅ + 30 = 2๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ
29. (3๐ฅ๐ฅ + 4)(3๐ฅ๐ฅ โ 4) = โ10๐ฅ๐ฅ 30. (5๐ฅ๐ฅ + 1)(๐ฅ๐ฅ + 3) = โ2(5๐ฅ๐ฅ + 1)
31. 4(๐ฅ๐ฅ โ 3)2 โ 36 = 0 32. 3(๐๐ + 5)2 โ 27 = 0
33. (๐ฅ๐ฅ โ 3)(๐ฅ๐ฅ + 5) = โ7 34. (๐ฅ๐ฅ + 8)(๐ฅ๐ฅ โ 2) = โ21
35. (2๐ฅ๐ฅ โ 1)(๐ฅ๐ฅ โ 3) = ๐ฅ๐ฅ2 โ ๐ฅ๐ฅ โ 2 36. 4๐ฅ๐ฅ2 + ๐ฅ๐ฅ โ 10 = (๐ฅ๐ฅ โ 2)(๐ฅ๐ฅ + 1)
37. 4(2๐ฅ๐ฅ + 3)2 โ (2๐ฅ๐ฅ + 3) โ 3 = 0 38. 5(3๐ฅ๐ฅ โ 1)2 + 3 = โ16(3๐ฅ๐ฅ โ 1)
39. ๐ฅ๐ฅ3 + 2๐ฅ๐ฅ2 โ 15๐ฅ๐ฅ = 0 40. 6๐ฅ๐ฅ3 โ 13๐ฅ๐ฅ2 โ 5๐ฅ๐ฅ = 0
41. 25๐ฅ๐ฅ3 = 64๐ฅ๐ฅ 42. 9๐ฅ๐ฅ3 = 49๐ฅ๐ฅ
43. ๐ฅ๐ฅ4 โ 26๐ฅ๐ฅ2 + 25 = 0 44. ๐๐4 โ 50๐๐2 + 49 = 0
45. ๐ฅ๐ฅ3 โ 6๐ฅ๐ฅ2 = โ8๐ฅ๐ฅ 46. ๐ฅ๐ฅ3 โ 2๐ฅ๐ฅ2 = 3๐ฅ๐ฅ
47. ๐๐3 + ๐๐2 โ 9๐๐ โ 9 = 0 48. 2๐ฅ๐ฅ3 โ ๐ฅ๐ฅ2 โ 2๐ฅ๐ฅ + 1 = 0
49. 5๐ฅ๐ฅ3 + 2๐ฅ๐ฅ2 โ 20๐ฅ๐ฅ โ 8 = 0 50. 2๐ฅ๐ฅ3 + 3๐ฅ๐ฅ2 โ 18๐ฅ๐ฅ โ 27 = 0 Discussion Point
51. A student tried to solve the equation ๐ฅ๐ฅ3 = 9๐ฅ๐ฅ by first dividing each side by ๐ฅ๐ฅ, obtaining ๐ฅ๐ฅ2 = 9. She then solved the resulting equation by the zero-product property and obtained the solution set {โ3,3}. Is this a complete solution? Explain your reasoning.
Analytic Skills
52. Given that ๐๐(๐ฅ๐ฅ) = ๐ฅ๐ฅ2 + 14๐ฅ๐ฅ + 50, find all values of ๐ฅ๐ฅ such that ๐๐(๐ฅ๐ฅ) = 5.
53. Given that ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2 โ 15๐ฅ๐ฅ, find all values of ๐ฅ๐ฅ such that ๐๐(๐ฅ๐ฅ) = โ7.
54. Given that ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2 + 3๐ฅ๐ฅ and ๐๐(๐ฅ๐ฅ) = โ6๐ฅ๐ฅ + 5, find all values of ๐ฅ๐ฅ such that ๐๐(๐ฅ๐ฅ) = ๐๐(๐ฅ๐ฅ).
55. Given that ๐๐(๐ฅ๐ฅ) = 2๐ฅ๐ฅ2 + 11๐ฅ๐ฅ โ 16 and โ(๐ฅ๐ฅ) = 5 + 2๐ฅ๐ฅ โ ๐ฅ๐ฅ2, find all values of ๐ฅ๐ฅ such that ๐๐(๐ฅ๐ฅ) = โ(๐ฅ๐ฅ).
Solve each equation for the specified variable.
56. ๐ท๐ท๐๐๐ก๐ก = ๐ด๐ด โ ๐ท๐ท, for ๐ท๐ท 57. 3๐๐ + 2๐๐ = 5 โ ๐๐๐๐, for ๐๐
58. 5๐๐ + ๐๐๐๐ = ๐๐ โ 2๐๐, for ๐๐ 59. ๐ธ๐ธ = ๐ ๐ +๐๐๐๐
, for ๐๐
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section F4 | 251
Solving Polynomial Equations and Applications of Factoring
60. ๐ง๐ง = ๐ฅ๐ฅ+2๐๐
๐๐, for ๐๐ 61. ๐๐ = โ2๐๐+4
๐๐, for ๐๐
Analytic Skills
Use factoring the GCF out to solve each formula for the indicated variable.
62. An object is thrown downwards, with an initial speed of 16 ft/s, from the top of a building 480 ft high. If the distance travelled by the object, in ft, is given by the function ๐๐(๐ก๐ก) = ๐ข๐ข๐ก๐ก + 16๐ก๐ก2, where ๐ข๐ข is the initial speed in ft/s, and ๐ก๐ก is the time in seconds, then how many seconds later will the object hit the ground?
63. A sandbag is dropped from a hot-air balloon 900 ft above the ground. The height, โ, of the sandbag above the ground, in feet, after ๐ก๐ก seconds is given by the function โ(๐ก๐ก) = 900 โ 16๐ก๐ก2. When will the sandbag hit the ground?
64. The sum of a number and its square is 72. Find the number.
65. The sum of a number and its square is 210. Find the number.
66. The length of a rectangle is 2 meters more than twice the width. The area of the rectangle is 84 m2. Find the length and width of the rectangle.
67. An envelope is 4 cm longer than it is wide. The area is 96 cm2. Find its length and width.
68. The height of a triangle is 8 cm more than the length of the base. The area of the triangle is 64 cm2. Find the base and height of the triangle.
69. A triangular sail is 9 m taller than it is wide. The area is 56 m2. Find the height and the base of the sail.
70. A gardener decides to build a stone pathway of uniform width around her flower bed. The flower bed measures 10 ft by 12 ft. If she wants the area of the bed and the pathway together to be 224 ft2, how wide should she make the pathway?
71. Suppose a rectangular flower bed is 3 m longer than it is wide. What are the dimensions of the flower bed if its area is 108 m2 ?
72. A picture frame measures 12 cm by 20 cm, and 84 cm2 of picture shows. Find the width of the frame.
73. A picture frame measures 14 cm by 20 cm, and 160 cm2 of picture shows. Find the width of the frame.
74. If each of the sides of a square is lengthened by 6 cm, the area becomes 144 cm2. Find the length of a side of the original square.
75. If each of the sides of a square is lengthened by 4 m, the area becomes 49 m2. Find the length of a side of the original square.
๐๐
๐๐ + 8
2๐ค๐ค + 2
๐ค๐ค