Polynomials
6-5
6-5 Multiplying
Multiplying
Polynomials
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra 1Algebra 1
Holt
McDougal
6-5 Multiplying Polynomials
Warm Up
Evaluate.
1. 32
2. 24 16
9
3. 102 100
Simplify.
4. 23  24 27
5. y5  y4 y9
6. (53)2
7. (x2)4
56
8. –4(x – 7)
Holt McDougal Algebra 1
–4x + 28
x8
6-5 Multiplying Polynomials
Objective
Multiply polynomials.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
To multiply monomials and polynomials,
you will use some of the properties of
exponents that you learned earlier in this
chapter.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 1: Multiplying Monomials
Multiply.
A. (6y3)(3y5)
(6y3)(3y5)
(6  3)(y3  y5)
18y8
Group factors with like bases
together.
Multiply.
B. (3mn2) (9m2n)
(3mn2)(9m2n)
(3  9)(m  m2)(n2  n)
27m3n3
Holt McDougal Algebra 1
Group factors with like bases
together.
Multiply.
6-5 Multiplying Polynomials
Example 1C: Multiplying Monomials
Multiply.
1 2 2
s
t
4
(st) (-12 s t2)
 1 2 2
2
s
t
t

12
t
s
(
)
s

÷
4


(
1
 2
 4 • −12÷s • s • s


(
Holt McDougal Algebra 1
)
Group factors with like
bases together.
)(t • t • t )
2
2
Multiply.
6-5 Multiplying Polynomials
Remember!
When multiplying powers with the same base,
keep the base and add the exponents.
x2  x3 = x2+3 = x5
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 1
Multiply.
a. (3x3)(6x2)
(3x3)(6x2)
(3  6)(x3  x2)
18x5
Group factors with like bases
together.
Multiply.
b. (2r2t)(5t3)
(2r2t)(5t3)
(2  5)(r2)(t3  t)
10r2t4
Holt McDougal Algebra 1
Group factors with like bases
together.
Multiply.
6-5 Multiplying Polynomials
Check It Out! Example 1 Continued
Multiply.
1 2 
3 2
4 5
x
y
12
x
z
yz
(
)(
c. 
÷
3

)
1 2 
3 2
x
y
12
x
z
÷
3


)(y z )
 2
1
3
•
12
x
x
•
3
÷


)( y •y )(z
(
(
4x 5y 5 z 7
Holt McDougal Algebra 1
4
5
4
2
• z5
)
Group factors with
like bases
together.
Multiply.
6-5 Multiplying Polynomials
To multiply a polynomial by a monomial, use
the Distributive Property.
Holt McDougal Algebra 1
6-5 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
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 2B: Multiplying a Polynomial by a Monomial
Multiply.
6pq(2p – q)
(6pq)(2p – q)
Distribute 6pq.
(6pq)2p + (6pq)(–q)
(6

2)(p

Group like bases
together.
p)(q) + (–1)(6)(p)(q  q)
12p2q – 6pq2
Holt McDougal Algebra 1
Multiply.
6-5 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
xy
x y 6
+ 8x y
2
(
1 2
Distribute x y .
2
)
1 2 
1 2 
2 2
Group like bases
x
y
6
xy
+
x
y
8
x
y
(
)

÷
2
÷

2


together.
 2
1  2
1
 • 6 ÷x • x ( y • y) +  • 8÷ x • x2 y • y2

2 
2
(
(
)
3x3y2 + 4x4y3
Holt McDougal Algebra 1
)
(
)(
Multiply.
)
6-5 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
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 2
Multiply.
b. 3ab(5a2 + b)
3ab(5a2 + b)
Distribute 3ab.
(3ab)(5a2) + (3ab)(b)
(3  5)(a  a2)(b) + (3)(a)(b  b)
15a3b + 3ab2
Holt McDougal Algebra 1
Group like bases
together.
Multiply.
6-5 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)
5r3s2 – 15r2s3
Holt McDougal Algebra 1
Group like bases
together.
Multiply.
6-5 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(x + 2) + 3(x + 2)
Distribute again.
= x(x) + x(2) + 3(x) + 3(2)
Multiply.
= x2 + 2x + 3x + 6
Combine like terms.
= x2 + 5x + 6
Holt McDougal Algebra 1
6-5 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)
I
3. Multiply the Inner terms. (x + 3)(x + 2)
L
4. Multiply the Last terms. (x + 3)(x + 2)
x  2 = 2x
3  x = 3x
3 2 = 6
(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
F
Holt McDougal Algebra 1
O
I
L
6-5 Multiplying Polynomials
Example 3A: Multiplying Binomials
Multiply.
(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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 3B: Multiplying Binomials
Multiply.
(x –
4)2
(x – 4)(x – 4)
Write as a product of
two binomials.
Use the FOIL method.
(x  x) + (x  (–4)) + (–4  x) + (–4  (–4))
x2 – 4x – 4x + 16
Multiply.
x2 – 8x + 16
Combine like terms.
Holt McDougal Algebra 1
6-5 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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Helpful Hint
In the expression (x + 5)2, the base is (x + 5).
(x + 5)2 = (x + 5)(x + 5)
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 3b
Multiply.
(x – 3)2
(x – 3)(x – 3)
Write as a product of
two binomials.
Use the FOIL method.
(x ●x) + (x(–3)) + (–3  x)+ (–3)(–3)
x2 – 3x – 3x + 9
Multiply.
x2 – 6x + 9
Combine like terms.
Holt McDougal Algebra 1
6-5 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.
Holt McDougal Algebra 1
6-5 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
Holt McDougal Algebra 1
6-5 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
5x
+3
10x3
6x2
+10x
–6
50x2 –30x
30x
–18
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
Holt McDougal Algebra 1
6-5 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.
Multiply each term in the top

5x + 3
polynomial by 5x, and align
6x2 + 30x – 18
like terms.
+ 10x3 + 50x2 – 30x
10x3 + 56x2 + 0x – 18 Combine like terms by adding
vertically.
10x3 + 56x2 +
– 18 Simplify.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 4A: Multiplying Polynomials
Multiply.
(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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 4B: Multiplying Polynomials
Multiply.
(2x – 5)(–4x2 – 10x + 3)
(2x – 5)(–4x2 – 10x + 3)
Multiply each term in the
top polynomial by –5.
–4x2 – 10x + 3
x
2x – 5
20x2 + 50x – 15
+ –8x3 – 20x2 + 6x
–8x3
+ 56x – 15
Multiply each term in the
top polynomial by 2x,
and align like terms.
Holt McDougal Algebra 1
Combine like terms by
adding vertically.
6-5 Multiplying Polynomials
Example 4C: Multiplying Polynomials
Multiply.
(x + 3)3
[x · x + x(3) + 3(x) + (3)(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)
Holt McDougal Algebra 1
Combine like terms.
6-5 Multiplying Polynomials
Example 4C: Multiplying Polynomials Continued
Multiply.
(x + 3)3
(x + 3)(x2 + 6x + 9)
Use the Commutative
Property of
Multiplication.
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
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example 4D: Multiplying Polynomials
Multiply.
(3x + 1)(x3 + 4x2 – 7)
–7
x3
–4x2
3x
3x4
–12x3 –21x
+1
x3
–4x2 –7
Write the product of the
monomials in each
row and column.
Add all terms inside the
rectangle.
3x4 – 12x3 + x3 – 4x2 – 21x – 7
3x4 – 11x3 – 4x2 – 21x – 7
Holt McDougal Algebra 1
Combine like terms.
6-5 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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Check It Out! Example 4b
Multiply.
(3x + 2)(x2 – 2x + 5)
(3x + 2)(x2 – 2x + 5)
x2 – 2x + 5

3x + 2
2x2 – 4x + 10
+ 3x3 – 6x2 + 15x
3x3 – 4x2 + 11x + 10
Holt McDougal Algebra 1
Multiply each term in the
top polynomial by 2.
Multiply each term in the
top polynomial by 3x,
and align like terms.
Combine like terms by
adding vertically.
6-5 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.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
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
A = h2 + h – 12
Write the formula for the 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.
Holt McDougal Algebra 1
6-5 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.
Write the formula for the
A = lw
area of a rectangle.
A = lw
A = x(x – 4)
A = x2 – 4x
Substitute x – 4 for l and
x for w.
Multiply.
The area is represented by x2 – 4x.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
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
A = x2 – 4x
meters shorter than width .
Substitute 6 for x.
A = 62 – 4  6
A = 36 – 24
Simplify.
A = 12
Combine terms.
The area is 12 square meters.
Holt McDougal Algebra 1
6-5 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
Holt McDougal Algebra 1
6-5 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 McDougal Algebra 1