1. Multiplying Polynomials
6-5
Multiplying Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Warm Up
Evaluate.
1. 32
3. 102
Simplify.
4. 23  24
6. (53)2 9
2. 24 16
100 27
5. y5  y4 y9 56
7. (x2)4 x8
8. –4(x – 7)
–4x + 28

3. Objective
Multiply polynomials.

4. To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

5. Example 1: Multiplying Monomials
A. (6y3)(3y5)
(6y3)(3y5)
Group factors with like bases together.
(6 3)(y3 y5) 
18y8
Multiply.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
Group factors with like bases together.
(3 9)(m m2)(n2  n) 
27m3n3
Multiply.

6. Example 1C: Multiplying Monomials4
s2 t2
(st) (-12 s t2) ( ) æ ç è - 2 1 12 4 t s ö ÷ ø
Group factors with like bases together. ( ) • æ ö ç è 2 1
−12 4 t s ÷ ø • •
Multiply.

7. When multiplying powers with the same base, keep the base and add the exponents.
x2  x3 = x2+3 = x5
Remember!

8. Group factors with like bases together. (3x3)(6x2)
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
Group factors with like bases together.
(3x3)(6x2)
(3 6)(x3 x2) 
Multiply.
18x5
b. (2r2t)(5t3)
Group factors with like bases together.
(2r2t)(5t3)
(2 5)(r2)(t3 t) 
Multiply.
10r2t4

9. Check It Out! Example 1 Continued
Multiply. æ 1 ö ( ) ( ) c.
x y 2 12
x z 3 2 4 5 ç ÷
y z è 3 ø ( ) æ ç è 4 5 2 1 12 3 x z y ö ÷ ø
Group factors with like bases together. ( ) æ ç è 3 2 4 5 1 12 z
x x
y y ö ÷ ø • •
Multiply. • • 7 5 4 x y z

10. To multiply a polynomial by a monomial, use the Distributive Property.

11. Example 2A: Multiplying a Polynomial by a Monomial
4(3x2 + 4x – 8)
Distribute 4.
4(3x2 + 4x – 8)
(4)3x2 +(4)4x – (4)8
Multiply.
12x2 + 16x – 32

12. Example 2B: Multiplying a Polynomial by a Monomial
6pq(2p – q)
Distribute 6pq.
(6pq)(2p – q)
(6pq)2p + (6pq)(–q)
Group like bases together.
(6  2)(p  p)(q) + (–1)(6)(p)(q  q)
12p2q – 6pq2
Multiply.

13. Example 2C: Multiplying a Polynomial by a Monomial1 ( )
x y 2 6 xy + 8
x y 2 2 2
Distribute 2 1
x y
x y ( ) + 2 6 1 xy y x 8
x y
x y ( ) æ ç è + 2 1 6 8 xy ö ÷ ø
Group like bases together.
x2 • x ( ) æ + ç è 1
• 6 2
y • y
x2 • x2
y • y2
• 8 ö ÷ ø
3x3y2 + 4x4y3
Multiply.

14. Check It Out! Example 2
Multiply.
a. 2(4x2 + x + 3)
Distribute 2.
2(4x2 + x + 3)
2(4x2) + 2(x) + 2(3)
Multiply.
8x2 + 2x + 6

15. Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
Distribute 3ab.
3ab(5a2 + b)
(3ab)(5a2) + (3ab)(b)
Group like bases together.
(3  5)(a  a2)(b) + (3)(a)(b  b)
15a3b + 3ab2
Multiply.

16. Check It Out! Example 2
Multiply.
c. 5r2s2(r – 3s)
Distribute 5r2s2.
5r2s2(r – 3s)
(5r2s2)(r) – (5r2s2)(3s)
Group like bases together.
(5)(r2  r)(s2) – (5  3)(r2)(s2  s)
5r3s2 – 15r2s3
Multiply.

17. To multiply a binomial by a binomial, you can apply the Distributive Property more than once:
Distribute.
(x + 3)(x + 2) = x(x + 2) + 3(x + 2)
Distribute again.
= x(x + 2) + 3(x + 2)
= x(x) + x(2) + 3(x) + 3(2)
Multiply.
= x2 + 2x + 3x + 6
Combine like terms.
= x2 + 5x + 6

18. Another method for multiplying binomials is called the FOIL method.
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) x = 3x  L
4. Multiply the Last terms. (x + 3)(x + 2) = 6 
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6 F O I L

19. ***Another method: Using table***

20. Example 3A: Multiplying Binomials
(s + 4)(s – 2)
(s + 4)(s – 2)
s(s – 2) + 4(s – 2)
Distribute.
s(s) + s(–2) + 4(s) + 4(–2)
Distribute again.
s2 – 2s + 4s – 8
Multiply.
s2 + 2s – 8
Combine like terms.

21. Example 3B: Multiplying Binomials
Write as a product of two binomials.
(x – 4)2
(x – 4)(x – 4)
Use the FOIL method.
(x x) + (x (–4)) + (–4  x) + (–4  (–4)) 
x2 – 4x – 4x + 16
Multiply.
x2 – 8x + 16
Combine like terms.

22. Example 3C: Multiplying Binomials
(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.

23. In the expression (x + 5)2, the base is (x + 5)
In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)
Helpful Hint

24. Check It Out! Example 3a
Multiply.
(a + 3)(a – 4)
(a + 3)(a – 4)
a(a – 4)+3(a – 4)
Distribute.
a(a) + a(–4) + 3(a) + 3(–4)
Distribute again.
a2 – 4a + 3a – 12
Multiply.
a2 – a – 12
Combine like terms.

25. Write as a product of two binomials. (x – 3)2
Check It Out! Example 3b
Multiply.
Write as a product of two binomials.
(x – 3)2
(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.

26. 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.

27. 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

28. 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
10x3
50x2
–30x
30x
6x2
–18 5x +3
Write the product of the monomials in each row and 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

29. Example 4A: Multiplying Polynomials
(x – 5)(x2 + 4x – 6)
(x – 5 )(x2 + 4x – 6)
Distribute x.
x(x2 + 4x – 6) – 5(x2 + 4x – 6)
Distribute x 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.

30. Example 4B: Multiplying Polynomials
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the top polynomial by –5.
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the top polynomial by 2x, and align like terms.
–4x2 – 10x + 3
2x – 5 x
20x2 + 50x – 15
+ –8x3 – 20x2 + 6x
Combine like terms by adding vertically.
–8x x – 15

31. Example 4C: Multiplying Polynomials
Write as the product of three binomials.
[x · x + x(3) + 3(x) + (3)(3)]
[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.

32. Example 4C: Multiplying Polynomials Continued
Use the Commutative Property of Multiplication.
(x + 3)(x2 + 6x + 9)
x(x2 + 6x + 9) + 3(x2 + 6x + 9)
Distribute.
x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)
Distribute again.
x3 + 6x2 + 9x + 3x2 + 18x + 27
Combine like terms.
x3 + 9x2 + 27x + 27

33. Example 4D: Multiplying Polynomials
(3x + 1)(x3 + 4x2 – 7)
Write the product of the monomials in each row and column. x3
–4x2 –7 3x
3x4
–12x3
–21x +1
–4x2 x3 –7
Add all terms inside the rectangle.
3x4 – 12x3 + x3 – 4x2 – 21x – 7
3x4 – 11x3 – 4x2 – 21x – 7
Combine like terms.

34. 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.
Helpful Hint 

35. Check It Out! Example 4a
Multiply.
(x + 3)(x2 – 4x + 6)
(x + 3 )(x2 – 4x + 6)
Distribute.
x(x2 – 4x + 6) + 3(x2 – 4x + 6)
Distribute 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.

36. Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
Multiply each term in the top polynomial by 2.
(3x + 2)(x2 – 2x + 5)
Multiply each term in the top polynomial by 3x, and align like terms.
x2 – 2x + 5
3x + 2 
2x2 – 4x + 10
+ 3x3 – 6x2 + 15x
Combine like terms by adding vertically.
3x3 – 4x2 + 11x + 10

37. Write the formula for the area of a rectangle.
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
A = l w 
Write the formula for the area of a rectangle.
Substitute h – 3 for w and h + 4 for l.
A = (h + 4)(h – 3)
A = h2 + 4h – 3h – 12
Multiply.
A = h2 + h – 12
Combine like terms.
The area is represented by h2 + h – 12.

38. Example 5: Application Continued
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 the base of the prism.
A = h2 + h – 12
A = – 12
Substitute 5 for h.
A = – 12
Simplify.
A = 18
Combine terms.
The area is 18 square feet.

39. The length of a rectangle is 4 meters shorter than its width.
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 
A = l w
Write the formula for the area of a rectangle.
Substitute x – 4 for l and x for w.
A = x(x – 4)
A = x2 – 4x
Multiply.
The area is represented by x2 – 4x.

40. Check It Out! Example 5 Continued
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 meters shorter than width .
A = x2 – 4x
A = 62 – 4  6
Substitute 6 for x.
A = 36 – 24
Simplify.
A = 12
Combine terms.
The area is 12 square meters.

41. Lesson Quiz: Part I
Multiply.
1. (6s2t2)(3st)
2. 4xy2(x + y)
3. (x + 2)(x – 8)
4. (2x – 7)(x2 + 3x – 4)
5. 6mn(m2 + 10mn – 2)
6. (2x – 5y)(3x + y)
18s3t3
4x2y2 + 4xy3
x2 – 6x – 16
2x3 – x2 – 29x + 28
6m3n + 60m2n2 – 12mn
6x2 – 13xy – 5y2

42. 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
h2 + 2h
b. Find the area when the height is 8 cm.
48 cm2