To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.

2. Like Terms
Like Terms refers to monomials that have the same variable(s) but may have
different coefficients. The variables in the terms must have the same powers.
Which terms are like? 3a2b, 4ab2, 3ab, -5ab2
4ab2 and -5ab2 are like.
Even though the others have the same variables, the exponents are not
the same.
3a2b = 3aab, which is different from 4ab2 = 4abb.

3. Constants are like terms.
Which terms are like? 2x, -3, 5b, 0
-3 and 0 are like.
Which terms are like? 3x, 2x2, 4, x
3x and x are like.
Which terms are like? 2wx, w, 3x, 4xw
2wx and 4xw are like.

4. Adding Polynomials
Add: (x2 + 3x + 1) + (4x2 +5)
Step 1: Underline like terms:
(x2 + 3x + 1) + (4x2 +5)
Notice: ‘3x’ doesn’t have a like term.
Step 2: Add the coefficients of like terms, do not change the powers of the
variables:
(x2 + 4x2) + 3x + (1 + 5)
5x2 + 3x + 6

5. Some people prefer to add polynomials by stacking them. If you choose to do
this, be sure to line up the like terms!
(x2 + 3x + 1)
(x2 + 3x + 1) + (4x2 +5) + (4x2 +5)
5x2 + 3x + 6
Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)
(2a2 + 3ab + 4b2)
(2a2+3ab+4b2) + (7a2+ab+-2b2) + (7a2 + ab + -2b2)
9a2 + 4ab + 2b2

6. • Add the following polynomials; you may stack them if you
prefer:
3 3 3
1) 3 x 7x 3x 4x 6x 3x
2 2 2
2) 2w w 5 4w 7w 1 6w 8w 4
3 2 3 3 2
3) 2a 3a 5a a 4a 3 3a 3a 9a 3

7. Subtracting Polynomials
Subtract: (3x2 + 2x + 7) - (x2 + x + 4)
Step 1: Change subtraction to addition (Keep-Change-Change.).
(3x2 + 2x + 7) + (- x2 + - x + - 4)
Step 2: Underline OR line up the like terms and add.
(3x2 + 2x + 7)
+ (- x2 + - x + - 4)
2x2 + x + 3

8. • Subtract the following polynomials by changing to
addition (Keep-Change-Change.), then add:
2 2 2
1) x x 4 3x 4x 1 2x 3x 5
2 2 2
2) 9y 3y 1 2y y 9 7y 4y 10
2 3 2 3 2
3) 2g g 9 g 3g 3 g g g 12

10. 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

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

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

13. Example 1C: Multiplying Monomials
Multiply.
1
s 2t 2 st 12 s t 2 Group factors with like
4 bases together.
1
12 s2 s s t2 t t2 Multiply.
4

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

15. 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

16. Check It Out! Example 1
Multiply.
1
3 2
c. x 2y 12 x z y4
z 5
3
1
x 2y 12 x 3z 2 y 4z 5 Group factors with like bases
3 together.
1
g 12 x2 g
x 3 y g
y 4 z 2 g z5 Multiply.
3
4 x 5y 5 z 7

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

18. 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

19. 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.

20. Example 2C: Multiplying a Polynomial by a Monomial
Multiply.
1
x 2y 6 xy 8 x2y 2
2
1 2 1 2.
x 2 y 6 xy 8 x2
y Distribute xy
2 2
1 1
x 2y 6 xy x2y 8 x 2y 2 Group like bases
2 2 together.
1 1
•6 x2 •x y•y •8 x2 • x2 y • y2
2 2
3x3y2 + 4x4y3 Multiply.

21. 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

22. 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.

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

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

25. 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

26. 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.

27. Example 3B: Multiplying Binomials
Multiply.
Write as a product of two
(x – 4)2 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.

28. 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.

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

30. 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.

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

32. 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.

33. 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

34. 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

35. 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.
Multiply each term in the top
2x2 + 10x – 6 polynomial by 3.
Multiply each term in the top
5x + 3 polynomial by 5x, and align
6x2 + 30x – 18 like terms.
+ 10x3 + 50x2 – 30x
Combine like terms by adding
10x3 + 56x2 + 0x – 18 vertically.
Simplify.
10x3 + 56x2 – 18

36. 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.

37. Example 4B: Multiplying Polynomials
Multiply.
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the top
(2x – 5)(–4x2 – 10x + 3) polynomial by –5.
Multiply each term in the
–4x2 – 10x + 3
top polynomial by 2x,
x 2x – 5 and align like terms.
20x2 + 50x – 15
+ –8x3 – 20x2 + 6x
Combine like terms by adding
–8x3 + 56x – 15 vertically.

38. 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.

39. 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) + 3(6x) + 3(9) Distribute the x and 3
again.
x3 + 6x2 + 9x + 3x2 + 18x + 27 Combine like terms.
x3 + 9x2 + 27x + 27

40. Example 4D: Multiplying Polynomials
Multiply.
(3x + 1)(x3 – 4x2 – 7)
Write the product of the
x3 4x2 –7 monomials in each
row and column.
3x 3x4 12x3 –21x
+1 x3 4x2 –7 Add all terms inside the
rectangle.
3x4 + 12x3 + x3 + 4x2 – 21x – 7
3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.

41. 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.

42. 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.

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

44. 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 and h + 4
A = (h + 4)(h – 3) for l.
A = h2 + 4h – 3h – 12 Multiply.
A = h2 + h – 12 Combine like terms.
The area is represented by h2 + h – 12.

45. 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
A = h2 + h – 12 area 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.

46. 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.

47. 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 meters
A = x2 – 4x 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.

48. Lesson Quiz: Part I
Multiply. 18s3t3
1. (6s2t2)(3st)
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

49. 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
h2 + 2h
2
b. Find the area when the height is 8 cm.
48 cm2