10.1 Adding and Subtracting Polynomials
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1 olynomial of two terms is a binomial. lynomial of three terms is a trinomial. 7xy 2 + 2y 8x 2 + 12xy + 2y 2 The constant term is 15. The degree is 3. The leading coefficient is 6. The leading coefficient of a polynomial is the coefficient of the variable with the largest exponent. 6x 3 – 2x 2 + 8x + 15 e constant term is the term without a variable. 10.1 Adding and Subtracting Polynomials
description
The degree is 3. The leading coefficient is 6. The constant term is 15. 10.1 Adding and Subtracting Polynomials. A polynomial of two terms is a binomial. 7 xy 2 + 2 y. 8 x 2 + 12 xy + 2 y 2. A polynomial of three terms is a trinomial. - PowerPoint PPT Presentation
Transcript of 10.1 Adding and Subtracting Polynomials
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A polynomial of two terms is a binomial.
A polynomial of three terms is a trinomial.
7xy2 + 2y
8x2 + 12xy + 2y2
The constant term is 15.
The degree is 3.
The leading coefficient is 6.
The leading coefficient of a polynomial is the coefficient of the
variable with the largest exponent.
6x3 – 2x2 + 8x + 15
The constant term is the term without a variable.
10.1 Adding and Subtracting Polynomials
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linear f (x) = mx + b one
f (x) = ax2 + bx + c, a 0quadratic two
cubic threef (x) = ax3 + bx2 + cx + d, a 0
DegreeFunction Equation
Common polynomial functions are named according to their degree.
The degree of a polynomial is the greatest of the degrees of any
of its terms. The degree of a term is the sum of the exponents of
the variables. Examples: 3y2 + 5x + 7
21x5y + 3x3 + 2y2
degree 2
degree 6
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To add polynomials, combine like terms.
Examples: Add (5x3 + 6x2 + 3) + (3x3 – 12x2 – 10).Use a horizontal format.
(5x3 + 6x2 + 3) + (3x3 – 12x2 – 10)= (5x3 + 3x3 ) + (6x2 – 12x2) + (3 – 10)
Rearrange and group like terms.
= 8x3 – 6x2 – 7 Combine like terms.
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Add (6x3 + 11x –21) + (2x3 + 10 – 3x) + (5x3 + x – 7x2 + 5).Use a vertical format.
6x3 + 11x – 212x3 – 3x + 10
5x3 – 7x2 + x + 5
13x3 – 7x2 + 9x – 6
Arrange terms of each polynomial in
descending order with like terms in
the same column.
Add the terms of each column.
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The additive inverse of the polynomial x2 + 3x + 2 is – (x2 + 3x + 2).
This is equivalent to the additive inverse of each of the terms.
– (x2 + 3x + 2) = – x2 – 3x – 2
To subtract two polynomials, add the additive inverse of the
second polynomial to the first.
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Example: Add (4x2 – 5xy + 2y2) – (– x2 + 2xy – y2).
(4x2 – 5xy + 2y2) – (– x2 + 2xy – y2)
= (4x2 – 5xy + 2y2) + (x2 – 2xy + y2)
= (4x2 + x2) + (– 5xy – 2xy) + (2y2 + y2)
= 5x2 – 7xy + 3y2
Rewrite the subtraction as the
addition of the additive inverse.
Rearrange and group like terms.
Combine like terms.
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Let P(x) = 2x2 – 3x + 1 and R(x) = – x3 + x + 5.
Examples: Find P(x) + R(x).P(x) + R(x) = (2x2 – 3x + 1) + (– x3 + x + 5)
= – x3 + 2x2 + (– 3x + x) + (1 + 5)= – x3 + 2x2 – 2x + 6
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To multiply a polynomial by a monomial,
use the distributive property and the rule for
multiplying exponential expressions.
Examples:. Multiply: 2x(3x2 + 2x – 1).
= 6x3 + 4x2 – 2x = 2x(3x2
) + 2x(2x) + 2x(–1)
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Multiply: – 3x2y(5x2 – 2xy + 7y2).
= – 3x2y(5x2 ) – 3x2y(– 2xy) – 3x2y(7y2)
= – 15x4y + 6x3y2 – 21x2y3
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To multiply two polynomials, apply the
distributive property.
Example: Multiply: (x – 1)(2x2 + 7x + 3).= (x – 1)(2x2) + (x – 1)(7x) + (x – 1)(3)= 2x3 – 2x2 + 7x2 – 7x + 3x – 3= 2x3 + 5x2 – 4x – 3
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Example: Multiply: (x – 1)(2x2 + 7x + 3).
Two polynomials can also be multiplied using a vertical
format.
– 2x2 – 7x – 3 2x3 + 7x2 + 3x
2x3 + 5x2 – 4x – 3
Multiply – 1(2x2 + 7x + 3).Multiply x(2x2 + 7x + 3).
Add the terms in each column.
2x2 + 7x + 3x – 1
Example:
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To multiply two binomials use a method called FOIL,
which is based on the distributive property. The letters
of FOIL stand for First, Outer, Inner, and Last.
1. Multiply the first terms.
3. Multiply the inner terms.
4. Multiply the last terms.
5. Add the products.2. Multiply the outer terms.
6. Combine like terms.
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Examples: Multiply: (2x + 1)(7x – 5).
= 2x(7x) + 2x(–5) + (1)(7x) + (1)(– 5)
= 14x2 – 10x + 7x – 5
= 14x2 – 3x – 5
First Outer Inner Last
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Multiply: (5x – 3y)(7x + 6y).
= 35x2 + 30xy – 21yx – 18y2
= 35x2 + 9xy – 18y2
= 5x(7x) + 5x(6y) + (– 3y)(7x) + (– 3y)(6y)First Outer Inner Last
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(a + b)(a – b)
= a2 – b2
The multiply the sum and difference of two terms,
use this pattern:
= a2 – ab + ab – b2
square of the first termsquare of the second term
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Examples: (3x + 2)(3x – 2)= (3x)2 – (2)2
= 9x2 – 4
(x + 1)(x – 1)= (x)2 – (1)2
= x2 – 1
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(a + b)2 = (a + b)(a + b)
= a2 + 2ab + b2
= a2 + ab + ab + b2
To square a binomial, use this pattern:
square of the first term
twice the product of the two terms square of the last term
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Examples: Multiply: (2x – 2)2 .= (2x)2 + 2(2x)(– 2) + (– 2)2
= 4x2 – 8x + 4
Multiply: (x + 3y)2 .= (x)2 + 2(x)(3y) + (3y)2
= x2 + 6xy + 9y2
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The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term.
Example: Factor 18x3 + 60x.
GCF = 6x18x3 + 60x = 6x (3x2) + 6x (10) Apply the distributive law
to factor the polynomial.
6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x
Check the answer by multiplication.
Find the GCF.
= 6x (3x2 + 10)
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Example: Factor 4x2 – 12x + 20.
Therefore, GCF = 4.
4x2 – 12x + 20 = 4x2 – 4 · 3x + 4 · 5
4(x2 – 3x + 5) = 4x2 – 12x + 20
Check the answer.= 4(x2 – 3x + 5)
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A common binomial factor can be factored out of certain expressions.
Example: Factor the expression 5(x + 1) – y(x + 1).
5(x + 1) – y(x + 1) = (5 – y)(x + 1)
(5 – y)(x + 1) = 5(x + 1) – y(x + 1)
Check.
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A difference of squares can be factored
using the formula
Example: Factor x2 – 9y2. = (x)2 – (3y)2
= (x + 3y)(x – 3y)
Write terms as perfect
squares.
Use the
formula.
a2 – b2 = (a + b)(a – b).
x2 – 9y2
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The same method can be used to factor any expression which can be written as a difference of squares.
Example: Factor 4(x + 1)2 – 25y 4.
= (2(x + 1))2 – (5y2)2
= [(2(x + 1)) + (5y2)][(2(x + 1)) – (5y2)]
= (2x + 2 + 5y2)(2x + 2 – 5y2)
4(x + 1)2 – 25y 4
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Some polynomials can be factored by grouping terms to produce a common binomial factor.
= (2x + 3)y – (2x + 3)2
= (2xy + 3y) – (4x + 6) Group terms.
Examples: Factor 2xy + 3y – 4x – 6.
Factor each pair of terms.
= (2x + 3)( y – 2) Factor out the common binomial.
2xy + 3y – 4x – 6
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Factor 2a2 + 3bc – 2ab – 3ac.
= 2a2 – 2ab + 3bc – 3ac
= (2a2 – 2ab) + (3bc – 3ac)
= 2a(a – b) + 3c(b – a)
Rearrange terms.
Group terms.
Factor.
= 2a(a – b) – 3c(a – b) b – a = – (a – b).
= (2a – 3c)(a – b) Factor.
2a2 + 3bc – 2ab – 3ac
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To factor a trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example,
x2 + 10x + 24 = (x + 4)(x + 6).
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One method of factoring trinomials is based on reversing the FOIL process.
Example: Factor x2 + 3x + 2. = (x + a)(x + b)
Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b.
= x2
FApply FOIL to multiply the binomials.
= x2 + (a + b) x + ab Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2.
= x2 + (1 + 2) x + 1 · 2
Therefore, x2 + 3x + 2 = (x + 1)(x + 2).
O I L + ax + bx + ab
x2 + 3x + 2
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Example: Factor x2 – 8x + 15. = (x + a)(x + b)
(x – 3)(x – 5) = x2 – 5x – 3x + 15
x2 – 8x + 15 = (x – 3)(x – 5).
Therefore a + b = – 8
Check:
= x2 + (a + b)x + ab
It follows that both a and b are negative.
= x2 – 8x + 15.
and ab = 15.
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Example: Factor x2 + 13x + 36. = (x + a)(x + b)
Check: (x + 4)(x + 9)
Therefore a and b are two positive factors of 36 whose sum is 13.
x2 + 13x + 36
= x2 + 9x + 4x + 36 = x2 + 13x + 36.
= (x + 4)(x + 9)
= x2 + (a + b) x + ab
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Example: Factor 4x3 – 40x2 + 100x.
= 4x(x2) – 4x(10x) + 4x(25) The GCF is 4x.
= 4x(x2 – 10x + 25) Use distributive property to factor out the GCF.
= 4x(x – 5)(x – 5) Factor the trinomial.
4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25)
= 4x(x2 – 10x + 25)
= 4x3 – 40x2 + 100x
4x3 – 40x2 + 100x
Check:
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Example: Factor 2x2 + 5x + 3.
= (2x + a)(x + b)
= 2x2 + (a + 2b)x + ab
For some a and b.
2x2 + 5x + 3 = (2x + 3)(x + 1)
Check: (2x + 3)(x + 1) = 2x2 + 2x + 3x + 3 = 2x2 + 5x + 3.
2x2 + 5x + 3
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Example: Factor 4x2 – 12x + 5.
This polynomial factors as (x + a)(4x + b) or (2x + a)(2x + b).
4x2 – 12x + 5 = (2x –1)(2x – 5)
The middle term –12x equals either (4a + b) x or (2a + 2b) x. Since a and b cannot both be positive, they must both be negative.
Since ab = + 5, a and b have the same sign.
a = –1, b = – 5 or a = 1 and b = 5.
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Trinomials which are quadratic in form are factored like quadratic trinomials.
Example: Factor 3x 4 + 28x2 + 9.
= 3u2 + 28u + 9
= (3x2 + 1)(x2 + 9)
= (3u + 1)(u + 9)
Let u = x2.
Factor.
Replace u by x2.
Many trinomials cannot be factored.
Example: Factor x2 + 3x + 5.
Let x2 + 3x + 5 = (x + a)(x + b) = x2 + (a + b) x + ab.
The trinomial x2 + 3x + 5 cannot be factored.
Then a + b = 3 and ab = 5. This is impossible.
3x 4 + 28x2 + 9
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Factor by Grouping Example 2: FACTOR: 6mx – 4m + 3rx – 2r Factor the first two terms: 6mx – 4m = 2m (3x - 2) Factor the last two terms: + 3rx – 2r = r (3x - 2) The green parentheses are the same so it’s the
common factor Now you have a common factor
(3x - 2) (2m + r)
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Example: The length of a rectangle is (x + 5) ft. The width
is (x – 6) ft. Find the area of the rectangle in terms of
the variable x.
A = L · W = Area
x – 6
x + 5
L = (x + 5) ft
W = (x – 6) ft
A = (x + 5)(x – 6 ) = x2 – 6x + 5x – 30
= x2 – x – 30
The area is (x2 – x – 30) ft2.
