Unit: Polynomials: Multiplying and...

Intro to Applied and PreCalc – Notes Unit: Polynomials and Factoring

Name ________________________

Unit: Polynomials:

Multiplying and Factoring

Specific

Outcome

10I.A.1 Demonstrate an understanding of factors of whole

numbers by determining:

Prime factors

Greatest common factor

Least common multiple

10I.A.3 Demonstrate an understanding of powers with

integral and rational exponents

10I.A.4 Demonstrate an understanding of the multiplication

of polynomial expressions

10I.A.5 Demonstrate an understanding of common factors

and trinomial factoring

Comments : ________________________________________________

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Review Lesson on Polynomials

Naming Polynomials Polynomials – by number of terms

Monomials (1 term)

Binomials (2 terms)

Trinomials (3 terms)

Polynomial with 4 terms

Polynomial with 5 terms

Etc.

Polynomials – by highest degree

Constant (no variable visible)

Linear (variable with exponent of 1)

Quadratic (variable with exponent of 2)

Cubic (variable with exponent of 3)

Quartic (variable with exponent of 4)

Quintic (variable with exponent of 5)

Sixth degree polynomial (variable with exponent of 6)

Etc.

Collecting and Combining “Like” Terms When collecting like terms, they must have the same variable and same exponent.

Add or subtract the coefficients (NOTE: the + and – signs belong to the term behind them).

Examples: Simplify the following

a) 6xy + 4x + 8xy b) 2x – 3y – 6x + 2y c) 3x + 5x2 – 7y – 8x + 8y


Adding and Subtracting Polynomials

Adding

1) Remove the brackets

2) Simplify by collecting like terms

3) Combine like terms

4) Arrange in alphabetical order and then from highest to lowest power

Example 1: Simplify (x2 + 4x + 3) + (2x2 + 5x + 1)

Subtracting

1) Remove the first set of brackets (unless there is a number in front of it other than +1).

NOTE: a subtraction sign is similar to -1.

2) Multiply the negative one through the second bracket (take the opposite for each term in

the second set of brackets) and remove the brackets.

3) Collect like terms

4) Combine like terms

Example 2: Simplify (4x2 – 5x + 7) – (3x2 + 2x – 4)


Multiplying and Dividing Monomials

Multiplying (coefficient coefficient)(variable variable)

remember to add exponents when multiplying powers

Examples: Find the product

a) (30x)(4y) b) -5x2(-3yz)

c) 4(3x – 2y + 7)

use the distributive property to expand each expression multiply each term inside the

brackets by the term outside the brackets

Dividing divide each term of the polynomial by the monomial (note example 3)

remember to subtract exponents for common variables

Examples: Simplify a) 12x

3

b) -18a3b5c7

6ab2c4

c) 8x5- 16x4 + 4x2 – 2x3

2x2

Assignment: Adding and Subtracting Polynomials handout

Multiplying and Dividing Monomials handout


Lesson 1: Multiplying Polynomials (Part 1)

Multiplying Monomial by Polynomial: Distribution Method

Examples: Multiply the following polynomials

a) 5y2(x2 - y) b) 4y(2y2 + 3y - 1)

Multiplying Binomial by Binomial: FOIL Method

A technique for multiplying two binomials is using the F.O.I.L. method.

The letters F. O. I. L. stand for , , ,

We always multiply these terms.

Steps :

1) Identify the first term in each bracket and them together.

2) Identify the most outside terms of the expression and multiply them together.

3) Identify the most inside terms of the expression and multiply them together.

4) Identify the last term in each bracket and multiply them together.

5) Simplify by collecting like terms.


c) (x +2)(x + 5)

First Outer Inner Last


Multiply the following polynomials

d) (x + 6)(x + 8) e) (2x - y)(3x + y)

f) (x - 2y)(x + 2y)

Binomial Squared:

g) (x + 5)2 *This is the same as (x + 5)(x + 5)

h) (2x - y)2


Assignment #1 FOIL

Multiply the following binomials using FOIL:

1) 3 4x x 2) 2 5x x

3) 2 1 3 2x x 4) 4 3 2x x

5) 4x 2

6) 3 3 4x x

7) 4 2 4x x 8) 5 4t t

9) 3 2 2 9w w 10) 2 2z z


11) a b 2 12) 5 5 6 1e e

13) 2 1x x 14) 5x 2

15) 7 3x y x y 16) 6 2x y x y

17) 4x y 2 18) 8 7x x

19) 3 4x y x y 20) 1 10x x

21) 8x y 2 22) 8x 2


Lesson 2: Multiplying Polynomials (Part 2)

Multiplying Binomial by Trinomial: Distribution Method


a) (y - 3)(y2 - 4y + 7)

b) (2x - 1)(2x2 + 5x - 3)

Examples: Expand or simplify the following polynomials

a) 3(x - 1)(2x - 3)

b) (5a + 4) + (a – 1)(a + 2) – (2a – 3)


Assignment #2 Distribution Method

Multiply the following polynomials

1) 23 2 5 2d d d 2) 24 5 9 1s s s

3) 23 6 4 7c c c 4) 25 7k k k

5) 2 25 2 2 6y y y y 6) 2 25 3 3 4 5r r r r

7) 2 22 4 2 7c c c c


Simplify the following polynomials:

1) 4 5 3 2 3 3 1y y y 2) 3 9 2 5 4 7 6 3a a a a

3) 2 3 5 6 5 4d e d e d e d e 4) 2

5 4 2 7 8 6n n n

5) 2 23 4 2 5 6w w w w 6) 2 4 5 2 3 5t s t s t s

7) 3 7 4 3 2 2a a a 8) 2

2 3 6 3b b b

9) 2 4 3x y x y x y x y 10) 2

4 6 2 3 2a c a c a c


Lesson 3: Greatest Common Factor and Prime Factors

The greatest common factor (GCF) is the largest factor shared by two or more terms. This

is the largest number that divides evenly into two or more numbers.

In order to determine the GCF of 2 or more numbers, you must begin by listing all the factors

of those numbers.

When a factor of a number has exactly two divisors, one and itself, the factor is a prime

factor.

For example,

The factors of 12 are 1, 2, 3, 4, 6, and 12.

The prime factors of 12 are 1, 2, and 3.

To determine the prime factorization of 12, write 12 as a product of its prime factors: 2 x 2

x 3, or 2² x 3

NOTE:

o The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

o Natural numbers greater than one that are not prime, are composite.

Example 1: Write the prime factorization of 3300. Begin by using a Factor Tree.


Example 2: List the factors of each of the following numbers. Then, identify the GCF.

a) 15 and 30

b) -24 and -48

Example 3: Determine the greatest common factor of 4xy and 2x²y.

Example 4: Determine the greatest common factor of the following sets of terms:

18x²yz, 27x²y²z, 9x²y²


Least/Lowest Common Multiple

The least/lowest common multiple (LCM) is the smallest multiple shared by two or more

terms. To generate multiples of a number, multiply the number by the natural numbers; that

is, 1, 2, 3, 4, 5, and so on.

For example, some multiples of 26 are:

26 · 1 = 26 26 · 2 = 52 26 · 3 = 78

For two or more natural numbers, we can determine their least common multiple.

Example 5: Determine the least common multiple of 18, 20, and 30.

Example 6: Mei is stacking toy blocks that are 12 cm tall next to blocks that are 18 cm tall. What is

the shortest height at which the two stacks will be the same height?


Assignment#3 Prime Factors, LCM’s and GCF’s

1. Two ropes are 48 m and 32 m long. Each rope is to be cut into equal pieces and all pieces

must have the same length that is a whole number of meters. What is the greatest possible

length of each piece?

a) List the factors of both numbers.

b) State the Greatest Common Factor (GCF).

2. Complete each factor tree and write the prime factorization for each number.

a) 144 b) 600

c) 252 d) 900


3. Find the GCF for the following:

a. 44 and 70

b. 36 and 48

4. Find the Lowest Common Multiple (LCM) of the following:

a. 12 and 30

b. 16 and 18

5. Hamburger patties come in packages of 8. Buns come in packages of 6. What is the least

number of hamburgers that can be made with no patties or buns leftover?


6. List the prime factors of the coefficients and the variable for each term and find the

Greatest Common Factor between the two terms.

a. 6x2, 12x

b. 20c2d3, 30cd2

c. 4b2c3 , 6bc2

d. 18xy2z, 24x2y3z2

e. 5m3n, 20mn2

7. Determine the GCF of the following sets of terms:

a. 14a, 21b b. -5n2, -10n

c. 3rs, 7t d. 12f2g3, 16fg2, 32f3g2

e. -15d2e3, -30cd2e, -45cde f. -18j3k, 27j2kl, 36j2k2l2


Lesson 4: Common Factoring

Factoring is the process of . The better you are at

multiplying, the better you will be at factoring.

Multiplication Factoring

5x(x – 2y) = 5x2 – 10xy 5x2 – 10xy = 5x(x – 2y)

(x – 3)(x + 5) = x2 + 2x – 15 x2 + 2x – 15 = (x – 3)(x + 5)

1) Common Factoring: When factoring, begin by looking for the GCF. It could contain a number, a

variable or both.

Place this greatest common factor in front of parentheses, with the remaining polynomial

the parentheses.

Once this is done, the same number of terms as in the original question should be inside. (i.e. a

leaves a .)

Examples: Factor the following:

i) 4x + 8 = ( ) ii) 8xy – 32y2 = GCF Remaining factor

iii) 7n2 – 49n = iv) 15w3 + 5w =

iv) b – b2r3c = vi) 12n3 – 16n2 + 32n =

vii) 3x3 – 6x2y + 9xy2 =

Factoring can always be quickly and easily checked by the polynomials

together to see if the product is the original polynomial.


Assignment#4 Factoring Binomials Factor the following polynomials.

a. 6s +30 b. 4t +28

c. 5a – 5 d. 216 12r r

e. 7 14 49xy xy xz f. 3 2 23 9 27c c d

f. 215 5w w h.

2 34 6a a

i. 2 210 50x y xy j.

22 4g g

k. 2 2 235 15 5x y x y xy l. 2 3 22 6 4r r s rs


Lesson 5: Factoring ax2 + bx + c (leading coefficient)

Use this method anytime there is a in front of your x2 which

cannot be factored out.

Product-Sum-Factor (PSF) Method

Factor: 3x2 + 17x + 10

ALWAYS begin factoring by checking for common

.

From the remaining trinomial, calculate the product of

the and coefficients.

List all of the of this product.

(List them as pairs)

From these pairs of factors, identify which pair, when

added together, result in the middle term of the trinomial.

Re-write the trinomial with the original first and last

terms. In between these two terms, insert two new

terms using the pair of coefficients from step .

Now the first two and the

last two terms. (Notice how a common factor emerges.)

Now again. Collect the common

brackets.

You can your answer by expanding

(multiplying using FOIL)


Examples: Factor the following fully, if possible:

1. 3y2 – 10y + 8 =

2. 8a2 + 18a – 5 =

3. 15x2 + 48x + 36=

4. 2c2 + 2c – 3 =

5. 5x2 – 20xy + 20y2 =


6. 6b4 + 7b2 – 10 =

7. 2d3 + 7d2 – 30d =

8. 10g2 – 3gh – h2 =

9. 6k2 + 14km – 12m2 =


Assignment #5 Factoring with leading coefficients. Factor the following trinomials:

1. 22 3 1x x 2. 23 5 2x x

3. 26 13 6x x 4. 22 5 12x x

5. 24 18 10x x 6. 23 17 28x x

7. 22 11 14r r 8. 22 11 12l l

9. 23 9 6w w 10. 210 8 2b b


11. 2 25 6y yz z 12. 212 19 4a a

13. 22 7 15f f 14. 2 110r r

15. 26 6 3b b 16. 2 210 17 3m mn n

17. 2 29 9 2g gf f 18. 26 32 42l l


Lesson 6: Trinomial Factoring (leading coefficient of 1)

Trinomials will factor to 2 brackets.

Example: x2 + 5x + 6

Steps: ALWAYS factor out any terms/variables first.

Identify the of the last term of the trinomial.

Next, determine which of factors either up to or to get

the middle term

Therefore, x2 + 5x + 6 factors to ( )( )

Examples: Factor the following trinomials fully, if possible:

1. x2 + 9x + 18 2. y2 – 2y - 15

3. 5 + b2 – 6b 4. a2 – 4a - 60


5. x2 + 8xy + 16y2 6. p4 –2p2 – 15

7. 2x2 + 8x + 6 8. x2 - 7x - 6

9. 3x3 - 18x2 + 27x 10. 2x2yz3 - 10xyz3 - 48z3y


Assignment #6 Factoring Trinomials

Factor the following trinomials:

1. 2 8 12y y 2. 2 10 21x x

3. 2 19 90a a 4. 2 242m mn n

5. 2 19 34b b 6. 2 10 24g g

7. 2 15 26n n 8. 2 15 56c c

9. 2 27 10s st t 10. 2 6 12f f


Lesson 7: Factoring Difference of Squares

A) Perfect Square Binomials: ax2 - by2

A difference of squares has 3 main features:

1. The first term is a perfect .

2. The second term is a square.

3. They are separated by a sign.

The term is absent because it is .

Eg. x2 - 0xy - 16y2

Factoring a perfect square binomial results in two similar binomials,

that differ only in the sign.

To factor a difference of squares:

Remember to ALWAYS begin factoring by looking for a factor.

The first term of the binomials’ comes from the square of the

term.

The term of the binomials’ comes from the square of the second

term.

Place a sign in one parentheses and a in the other.

Check the result by using F.O.I.L .

Example from above: x2 - 16y2 = ( )( )

Examples:

1. x2 - 9

2. 225b2 - a2

3. 49 + x2

4. -y2 + 36

5. 3x3 - 48x

6. x4 - 16

Example:

x2 – 16y

2


B) Perfect Square Trinomials: x2 bxy + cy2

A perfect square trinomial has main features:

1. The first term is a .

2. The term is a perfect square. The sign of the last term

is always .

3. The term can be either positive or negative. It is always double the

square root of the last term.

Factoring a perfect square trinomial results in two

Example: Factor: x2 - 8xy + 16y2

Check:

Examples: Factor the following trinomials fully, if possible.

1. 49 + 14x +x2

2. 5b3 - 40b2 + 80b

3. The volume of a rectangular prism is represented by 2x3 - 24x2 + 72x. What are possible

dimensions of the prism? (V = LxWxH)

Example:

x2 – 8xy + 16y

2


Assignment #7 Factoring Difference of Squares and Perfect

Squares

Factor the following DOS’s and POS’s:

1. 2 5 25n n 2. 2 100a

3. 2 49t 4. 2 4x

5. 264 h 6. 2 29 16c d

7. 2 2r s 8. 2 250 72g h

9. 2 29 15p r 10. 2 144s

11. 2 272 32g h 12. 2 12 36y y

13. 2 6 9x x 14. 22 12 18z z

15. 2 5 25a a 16. 2144 48 4b b


17. 2 216 64d e 18. 227 48m

19. 22 24 72k k 20. 3 23 51 147c c c

21. 2 2100 25a b 22. 3 218 81s t s t st

23. 481 16d 24. 2 212 12 3lnlm lmn


ANSWER KEY:

Assignment #1

1. 2. 3. 4.

5. 6. 7. 8. 20

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 22.

Assignment #2

1. 2. 3.

4. 5.

6. 7.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

Assignment #3

1. GCF = 16

2. a) b) c) d)

3. a) GCF = 2 b) GCF = 12 4. a) LCM = 60 b) LCM = 144

5. LCM = 24

6. a) GCF = 6x b) GCF = 10cd2 c) GCF = 2bc2 d) GCF = 6xy2z e) GCF = 5mn

7. a) GCF = 7 b) GCF = -5n c) GCF = 1 d) GCF = 4fg2 e) GCF = -15de

f) GCF = 9j2k


Assignment #4

a) 6(s + 5) b) 4(t + 7) c) 5(a – 1) d) 4r(4r – 3)

e) 7x(y + 2y – 7z) f) g) 5w(3w -1) h)

i) 10xy(xy – 5) j) 2g(g + 2) k) 5xy(7x + 3xy + 1) l) )

Assignment #5

1. (2x + 1)(x + 1) 2. (3x – 1)(x + 2) 3. (3x – 2)(2x – 3)

4. (2x – 3)(x + 4) 5. 2 (x – 5)(x + 1) 6. (3x – 4)(x + 7)

7. (2r + 7)(r + 2) 8. (2l + 3)(l + 4) 9. 3(w + 2)(w + 1)

10. 2(5b2 + 4b + 1) 11. (y + 3z)(y + 2z) 12. (4a + 1)(3a + 4)

13. (2f – 3)(f + 5) 14. (r + 11)(r – 10) 15. 3(2b2 + 2b – 1)

16. (2m – 3n)(5m – n) 17. (3g – 2f)(3g – f) 18. 2(3l + 7)(l + 3)

Assignment #6

1. (y + 6)(y + 2) 2. (x + 7)(x + 3) 3. (a – 10)(a – 9)

4. (m – 7n)(m + 6n) 5. (b + 2)(b + 17) 6. (g – 6)(g – 4)

7. (n – 13)(n – 2) 8. (c – 8)(c – 7) 9. (s – 5t)(s – 2t)

10. Not Factorable

Assignment #7

1. not factorable 2. (a – 10)(a + 10) 3. (t – 7)(t + 7)

4. Not factorable 5. (8 – h)(8 + h) 6. (3c – 4d)(3c + 4d)

7. (r – s)(r + s) 8. 2(5g – 6h)(5g + 6h) 9. 3(3p2 – 5r2)

10. Not Factorable 11. 8(3g – 2h)(3g + 2h) 12. (y + 6)2

13. (x – 3)2 14. 2(z + 3)2 15. Not Factorable

16. 4(6 – b)2

Unit: Polynomials: Multiplying and...

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