This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.

1. 7-7 Multiplying Polynomials
7-7 Multiplying Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 1 1
Holt Algebra

2. 7-7 Multiplying Polynomials
Warm Up
Evaluate.
1. 32 9 2. 24 16
3. 102 100
Simplify.
4. 23 • 24 27 5. y5 • y4 y9
6. (53)2 56 7. (x2)4 x8
8. –4(x – 7) –4x + 28
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3. 7-7 Multiplying Polynomials
Objective
Multiply polynomials.
Holt Algebra 1

4. 7-7 Multiplying Polynomials
To multiply monomials and polynomials,
you will use some of the properties of
exponents that you learned earlier in this
chapter.
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5. 7-7 Multiplying Polynomials
Example 1: Multiplying Monomials
Multiply.
A. (6y3)(3y5)
(6y3)(3y5) Group factors with like bases
(6 • 3)(y3 • y5) together.
18y8 Multiply.
B. (3mn2) (9m2n)
(3mn2)(9m2n) Group factors with like bases
(3 • 9)(m • m2)(n2 • n) together.
27m3n3 Multiply.
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6. 7-7 Multiplying Polynomials
Example 1C: Multiplying Monomials
Multiply.
⎛ 1 2 2 ⎞
⎜ s t ⎟ (s t ) - 12 st 2
( ) Group factors with like
⎝ 4 ⎠ bases together.
⎛ 1 ⎞
⎜ 4 i - 12 ⎟ s 2 i s i s
( )(t 2 i t i t2 ) Multiply.
⎝ ⎠
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7. 7-7 Multiplying Polynomials
Remember!
When multiplying powers with the same base,
keep the base and add the exponents.
x2 • x3 = x2+3 = x5
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8. 7-7 Multiplying Polynomials
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
(3x3)(6x2) Group factors with like bases
together.
(3 • 6)(x3 • x2)
Multiply.
18x5
b. (2r2t)(5t3)
Group factors with like bases
(2r2t)(5t3) together.
(2 • 5)(r2)(t3 • t) Multiply.
10r2t4
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9. 7-7 Multiplying Polynomials
Check It Out! Example 1
Multiply.
⎛ 1 2 ⎞ 3 2 4 5
c. ⎜ x y (12 x z )( y z
⎟ )
⎝ 3 ⎠
⎛ 1 2 ⎞
⎜ 3 x y⎟ 12 x 3z 2
( )(y 4 z5 ) Group factors with
⎝ ⎠ like bases
together.
⎛ 1 i ⎞ 2 i 3
⎜ 3 12⎟ x x
⎝ ⎠
( )(y i y )(z4 2 i z5 ) Multiply.
4 x 5y 5 z 7
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10. 7-7 Multiplying Polynomials
To multiply a polynomial by a monomial, use
the Distributive Property.
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11. 7-7 Multiplying Polynomials
Example 2A: Multiplying a Polynomial by a Monomial
Multiply.
4(3x2 + 4x – 8)
4(3x2 + 4x – 8) Distribute 4.
(4)3x2 +(4)4x – (4)8 Multiply.
12x2 + 16x – 32
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12. 7-7 Multiplying Polynomials
Example 2B: Multiplying a Polynomial by a Monomial
Multiply.
6pq(2p – q)
(6pq)(2p – q) Distribute 6pq.
(6pq)2p + (6pq)(–q) Group like bases
together.
(6 • 2)(p • p)(q) + (–1)(6)(p)(q • q)
12p2q – 6pq2 Multiply.
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13. 7-7 Multiplying Polynomials
Example 2C: Multiplying a Polynomial by a Monomial
Multiply.
1 2 2 2
x y(6xy + 8 x y )
2
1 2 2 2 1 2
x y 6xy + 8 x y
( ) Distribute x y .
2 2
⎛ 1 2 ⎞ ⎛ 1 2 ⎞
⎜ 2 x y⎟ (6 xy ) + ⎜ x y ⎟ 8x 2 y2
( ) Group like bases
⎝ ⎠ ⎝ 2 ⎠ together.
⎛ 1 ⎞ 2 ⎛ 1 ⎞
⎜ • 6 ⎟ x • x ( y • y) + ⎜ • 8⎟ x2 • x2 y • y2
( ) ( )( )
⎝ 2 ⎠ ⎝ 2 ⎠
3x3y2 + 4x4y3 Multiply.
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14. 7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
a. 2(4x2 + x + 3)
2(4x2 + x + 3) Distribute 2.
2(4x2) + 2(x) + 2(3) Multiply.
8x2 + 2x + 6
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15. 7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
Distribute 3ab.
(3ab)(5a2) + (3ab)(b)
Group like bases
(3 • 5)(a • a2)(b) + (3)(a)(b • b) together.
15a3b + 3ab2 Multiply.
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16. 7-7 Multiplying Polynomials
Check It Out! Example 2
Multiply.
c. 5r2s2(r – 3s)
5r2s2(r – 3s) Distribute 5r2s2.
(5r2s2)(r) – (5r2s2)(3s)
(5)(r2 • r)(s2) – (5 • 3)(r2)(s2 • s) Group like bases
together.
5r3s2 – 15r2s3 Multiply.
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17. 7-7 Multiplying Polynomials
To multiply a binomial by a binomial, you can
apply the Distributive Property more than once:
(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.
= x(x + 2) + 3(x + 2) Distribute x and 3
again.
= x(x) + x(2) + 3(x) + 3(2) Multiply.
= x2 + 2x + 3x + 6 Combine like terms.
= x2 + 5x + 6
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18. 7-7 Multiplying Polynomials
Another method for multiplying binomials is
called the FOIL method.
F
1. Multiply the First terms. (x + 3)(x + 2) x • x = x2
O
2. Multiply the Outer terms. (x + 3)(x + 2) x • 2 = 2x
I
3. Multiply the Inner terms. (x + 3)(x + 2) 3 • x = 3x
L
4. Multiply the Last terms. (x + 3)(x + 2) 3• 2 = 6
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F O I L
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19. 7-7 Multiplying Polynomials
Example 3A: Multiplying Binomials
Multiply.
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2) Distribute s and 4.
s(s) + s(–2) + 4(s) + 4(–2) Distribute s and 4
again.
s2 – 2s + 4s – 8 Multiply.
s2 + 2s – 8 Combine like terms.
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20. 7-7 Multiplying Polynomials
Example 3B: Multiplying Binomials
Multiply.
Write as a product of
(x – 4)2
two binomials.
(x – 4)(x – 4) Use the FOIL method.
(x • x) + (x • (–4)) + (–4 • x) + (–4 • (–4))
x2 – 4x – 4x + 8 Multiply.
x2 – 8x + 8 Combine like terms.
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21. 7-7 Multiplying Polynomials
Example 3C: Multiplying Binomials
Multiply.
(8m2 – n)(m2 – 3n) Use the FOIL method.
8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)
8m4 – 24m2n – m2n + 3n2 Multiply.
8m4 – 25m2n + 3n2 Combine like terms.
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22. 7-7 Multiplying Polynomials
Helpful Hint
In the expression (x + 5)2, the base is (x + 5). (x
+ 5)2 = (x + 5)(x + 5)
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23. 7-7 Multiplying Polynomials
Check It Out! Example 3a
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4) Distribute a and 3.
a(a – 4)+3(a – 4) Distribute a and 3
again.
a(a) + a(–4) + 3(a) + 3(–4)
a2 – 4a + 3a – 12 Multiply.
a2 – a – 12 Combine like terms.
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24. 7-7 Multiplying Polynomials
Check It Out! Example 3b
Multiply.
Write as a product of
(x – 3)2
two binomials.
(x – 3)(x – 3) Use the FOIL method.
(x ● x) + (x•(–3)) + (–3 • x)+ (–3)(–3)
x2 – 3x – 3x + 9 Multiply.
x2 – 6x + 9 Combine like terms.
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25. 7-7 Multiplying Polynomials
Check It Out! Example 3c
Multiply.
(2a – b2)(a + 4b2)
(2a – b2)(a + 4b2) Use the FOIL method.
2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)
2a2 + 8ab2 – ab2 – 4b4 Multiply.
2a2 + 7ab2 – 4b4 Combine like terms.
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26. 7-7 Multiplying Polynomials
To multiply polynomials with more than two terms,
you can use the Distributive Property several times.
Multiply (5x + 3) by (2x2 + 10x – 6):
(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)
= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)
= 10x3 + 50x2 – 30x + 6x2 + 30x – 18
= 10x3 + 56x2 – 18
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27. 7-7 Multiplying Polynomials
You can also use a rectangle model to multiply
polynomials with more than two terms. This is
similar to finding the area of a rectangle with
length (2x2 + 10x – 6) and width (5x + 3):
2x2 +10x –6
Write the product of the
5x 10x3 50x2 –30x
monomials in each row and
+3 6x2 30x –18 column:
To find the product, add all of the terms inside the
rectangle by combining like terms and simplifying
if necessary.
10x3 + 6x2 + 50x2 + 30x – 30x – 18
10x3 + 56x2 – 18
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28. 7-7 Multiplying Polynomials
Another method that can be used to multiply
polynomials with more than two terms is the
vertical method. This is similar to methods used to
multiply whole numbers.
2x2 + 10x – 6 Multiply each term in the top
polynomial by 3.
× 5x + 3 Multiply each term in the top
6x2 + 30x – 18 polynomial by 5x, and align
+ 10x3 + 50x2 – 30x like terms.
10x3 + 56x2 + 0x – 18 Combine like terms by adding
vertically.
10x3 + 56x2 + – 18 Simplify.
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29. 7-7 Multiplying Polynomials
Example 4A: Multiplying Polynomials
Multiply.
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6) Distribute x and –5.
x(x2 + 4x – 6) – 5(x2 + 4x – 6) Distribute x and −5
again.
x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)
x3 + 4x2 – 5x2 – 6x – 20x + 30 Simplify.
x3 – x2 – 26x + 30 Combine like terms.
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30. 7-7 Multiplying Polynomials
Example 4B: Multiplying Polynomials
Multiply.
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the
(2x – 5)(–4x2 – 10x + 3) top polynomial by –5.
–4x2 – 10x + 3 Multiply each term in the
x 2x – 5 top polynomial by 2x,
20x2 + 50x – 15 and align like terms.
+ –8x3 – 20x2 + 6x
Combine like terms by
–8x3 + 56x – 15 adding vertically.
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31. 7-7 Multiplying Polynomials
Example 4C: Multiplying Polynomials
Multiply.
(x + 3)3
[(x + 3)(x + 3)](x + 3) Write as the product of
three binomials.
[x(x+3) + 3(x+3)](x + 3) Use the FOIL method on
the first two factors.
(x2 + 3x + 3x + 9)(x + 3) Multiply.
(x2 + 6x + 9)(x + 3) Combine like terms.
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32. 7-7 Multiplying Polynomials
Example 4C: Multiplying Polynomials
Multiply.
(x + 3)3 Use the Commutative
Property of
(x + 3)(x2 + 6x + 9)
Multiplication.
x(x2 + 6x + 9) + 3(x2 + 6x + 9) Distribute the x and 3.
x(x2) + x(6x) + x(9) + 3(x2) + Distribute the x and 3
3(6x) + 3(9) again.
x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms.
x3 + 9x2 + 27x + 27
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33. 7-7 Multiplying Polynomials
Example 4D: Multiplying Polynomials
Multiply.
(3x + 1)(x3 – 4x2 – 7)
Write the product of the
x3 4x2 –7
monomials in each
3x 3x4 12x3 –21x row and column.
+1 x3 4x2 –7 Add all terms inside the
rectangle.
3x4 + 12x3 + x3 + 4x2 – 21x – 7
3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.
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34. 7-7 Multiplying Polynomials
Helpful Hint
A polynomial with m terms multiplied by a
polynomial with n terms has a product that,
before simplifying has mn terms. In Example 4A,
there are 2 • 3, or 6 terms before simplifying.
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35. 7-7 Multiplying Polynomials
Check It Out! Example 4a
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6) Distribute x and 3.
x(x2 – 4x + 6) + 3(x2 – 4x + 6) Distribute x and 3
again.
x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)
x3 – 4x2 + 3x2 +6x – 12x + 18 Simplify.
x3 – x2 – 6x + 18 Combine like terms.
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36. 7-7 Multiplying Polynomials
Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
Multiply each term in the
(3x + 2)(x2 – 2x + 5) top polynomial by 2.
x2 – 2x + 5 Multiply each term in the
× 3x + 2 top polynomial by 3x,
2x2 – 4x + 10 and align like terms.
+ 3x3 – 6x2 + 15x
Combine like terms by
3x3 – 4x2 + 11x + 10 adding vertically.
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37. 7-7 Multiplying Polynomials
Example 5: Application
The width of a rectangular prism is 3 feet less
than the height, and the length of the prism is
4 feet more than the height.
a. Write a polynomial that represents the area of the
base of the prism.
A = l•w Write the formula for the
area of a rectangle.
A = l•w
Substitute h – 3 for w
A = (h + 4)(h – 3) and h + 4 for l.
A = h2 + 4h – 3h – 12 Multiply.
A = h2 + h – 12 Combine like terms.
The area is represented by h2 + h – 12.
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38. 7-7 Multiplying Polynomials
Example 5: Application
The width of a rectangular prism is 3 feet less
than the height, and the length of the prism is
4 feet more than the height.
b. Find the area of the base when the height is 5 ft.
A = h2 + h – 12
Write the formula for the area
A = h2 + h – 12 the base of the prism.
A = 52 + 5 – 12 Substitute 5 for h.
A = 25 + 5 – 12 Simplify.
A = 18 Combine terms.
The area is 18 square feet.
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39. 7-7 Multiplying Polynomials
Check It Out! Example 5
The length of a rectangle is 4 meters shorter
than its width.
a. Write a polynomial that represents the area of the
rectangle.
A = l•w Write the formula for the
area of a rectangle.
A = l•w
A = x(x – 4) Substitute x – 4 for l and
x for w.
A = x2 – 4x Multiply.
The area is represented by x2 – 4x.
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40. 7-7 Multiplying Polynomials
Check It Out! Example 5
The length of a rectangle is 4 meters shorter
than its width.
b. Find the area of a rectangle when the width is 6
meters.
A = x2 – 4x Write the formula for the area of a
rectangle whose length is 4
A = x2 – 4x
meters shorter than width .
A = 62 – 4 • 6 Substitute 6 for x.
A = 36 – 24 Simplify.
A = 12 Combine terms.
The area is 12 square meters.
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41. 7-7 Multiplying Polynomials
Lesson Quiz: Part I
Multiply.
1. (6s2t2)(3st) 18s3t3
2. 4xy2(x + y) 4x2y2 + 4xy3
3. (x + 2)(x – 8) x2 – 6x – 16
4. (2x – 7)(x2 + 3x – 4) 2x3 – x2 – 29x + 28
5. 6mn(m2 + 10mn – 2) 6m3n + 60m2n2 – 12mn
6. (2x – 5y)(3x + y) 6x2 – 13xy – 5y2
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42. 7-7 Multiplying Polynomials
Lesson Quiz: Part II
7. A triangle has a base that is 4cm longer than its
height.
a. Write a polynomial that represents the area
of the triangle.
1 2
h + 2h
2
b. Find the area when the height is 8 cm.
48 cm2
Holt Algebra 1