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
3. 102
Simplify.
4. 23  24
5. y5  y4
6. (53)2
7. (x2)4
8. –4(x – 7)
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Objective
Multiply polynomials.
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Review
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
Examples Together
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
Try these yourself!!!
Multiply.
a. (2r2t)(5t3)
1 2 
3 2
4 5
x
y
12
x
z
yz
(
)(
b. 

3

Holt McDougal Algebra 1
)
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
Try Yourself!!!
Multiply.
a. 3ab(5a2 + b)
b. 5r2s2(r – 3s)
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Method 1: To multiply a binomial by a binomial,
Distribute more than once (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
Try Yourself!!!
Multiply.
1. (a + 3)(a – 4)
3. (x – 3)2
Holt McDougal Algebra 1
2. (2a – b2)(a + 4b2)
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
Example
Multiply.
(x – 5)(x2 + 4x – 6)
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Example
Multiply.
(x + 3)3
Holt McDougal Algebra 1
6-5 Multiplying Polynomials
Method 2: Box Method
(2x2 + 10x – 6) and width (5x + 3):
2x2
+10x
5x
+3
–6
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
Example: Box Method
Multiply.
(3x + 1)(x3 + 4x2 – 7)
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
HOMEWORK
PG. 427
# 26-62 (Evens), 87-89
Holt McDougal Algebra 1