7.3
Multiplying Polynomials
How can you multiply two binomials?
1
ACTIVITY: Multiplying Binomials Using Algebra Tiles
Work with a partner. Six different algebra tiles are shown below.
1
Ź1
x
Źx
x2
Źx 2
Write the product of the two binomials shown by the algebra tiles.
a. (x + 3)(x − 2) =
b. (2x − 1)(2x + 1) =
c. (x + 2)(2x − 1) =
d. (−x − 2)(x − 3) =
COMMON
CORE
Polynomials
In this lesson, you will
● multiply binomials using
the Distributive Property, a
table, or the FOIL method.
● multiply binomials
and trinomials.
Learning Standard
A.APR.1
340
Chapter 7
Polynomial Equations and Factoring
2
ACTIVITY: Multiplying Monomials Using Algebra Tiles
Work with a partner. Write each product. Explain your reasoning.
Math
Practice
Use a Diagram
How can you
represent the
product of
polynomials using
diagrams?
3
a.
ä
b.
c.
ä
d.
ä
f.
ä
e.
ä
g.
ä
h.
ä
i.
ä
j.
ä
ä
ACTIVITY: Multiplying Binomials Using Algebra Tiles
Use algebra tiles to find each product.
a. (2x − 2)(2x + 1)
b. (4x + 3)(x − 2)
c. (−x + 2)(2x + 2)
d. (2x − 3)(x + 4)
e. (3x + 2)(−x − 1)
f. (2x + 1)(−3x + 2)
g. (x − 2)2
h. (2x − 3)2
4. IN YOUR OWN WORDS How can you multiply two binomials? Use the
results of Activity 3 to summarize a procedure for multiplying binomials
without using algebra tiles.
5. Find two binomials with the given product.
a. x 2 − 3x + 2
b. x 2 − 4x + 4
Use what you learned about multiplying binomials to complete
Exercises 3 and 4 on page 345.
Section 7.3
Multiplying Polynomials
341
7.3
Lesson
Lesson Tutorials
In Section 1.2, you used the Distributive Property to multiply a
binomial by a monomial. You can also use the Distributive Property
to multiply two binomials.
Key Vocabulary
FOIL Method, p. 343
EXAMPLE
1
Multiplying Binomials Using the Distributive Property
Find each product.
a. (x + 2)(x + 5)
Use the horizontal method.
Distribute (x + 5) to
each term of (x + 2).
(x + 2)(x + 5) = x(x + 5) + 2(x + 5)
= x(x) + x(5) + 2(x) + 2(5)
2
Distributive Property
= x + 5x + 2x + 10
Multiply.
= x 2 + 7x + 10
Combine like terms.
b. (x + 3)(x − 4)
Use the vertical method.
Multiply −4(x + 3).
Multiply x(x + 3).
x+3
x−4
×
−4x − 12
x2 + 3x
2
x − x − 12
Align like terms vertically.
Distributive Property
Distributive Property
Combine like terms.
The product is x 2 − x − 12.
EXAMPLE
2
Multiplying Binomials Using a Table
Find (2x − 3)(x + 5).
Step 1: Write each binomial as a sum of terms.
(2x − 3)(x + 5) = [2x + (−3)](x + 5)
2x
Step 2: Make a table of products.
The product is 2x 2 − 3x + 10x − 15,
or 2x 2 + 7x − 15.
x
2x
−3x
5
10x
−15
Use the Distributive Property to find the product.
Exercises 5–13
and 16–21
1. ( y + 4)( y + 1)
2.
(z − 2)(z + 6)
4.
(r − 5)(2r − 1)
Use a table to find the product.
3. ( p + 3)( p − 8)
342
Chapter 7
Polynomial Equations and Factoring
−3
2
The FOIL Method is a shortcut for multiplying two binomials.
FOIL Method
To multiply two binomials using the FOIL Method, find the sum of the
products of the
First terms,
(x + 1)(x + 2)
x(x) = x 2
Outer terms,
(x + 1)(x + 2)
x(2) = 2x
Inner terms, and
(x + 1)(x + 2)
1(x) = x
Last terms.
(x + 1)(x + 2)
1(2) = 2
(x + 1)(x + 2) = x 2 + 2x + x + 2 = x2 + 3x + 2
EXAMPLE
3
Multiplying Binomials Using the FOIL Method
Find each product.
a. (x − 3)(x − 6)
First
Outer
Inner
Last
(x − 3)(x − 6) = x(x) + x(−6) + (−3)(x) + (−3)(−6)
Use the FOIL Method.
= x 2 + (−6x) + (−3x) + 18
Multiply.
= x 2 − 9x + 18
Combine like terms.
b. (2x + 1)(3x − 5)
First
Outer
Inner
Last
(2x + 1)(3x − 5) = 2x(3x) + 2x(−5) + 1(3x) + 1(−5)
Use the FOIL Method.
= 6x 2 + (−10x) + 3x + (−5)
Multiply.
= 6x 2 − 7x − 5
Combine like terms.
Use the FOIL Method to find the product.
Exercises 22–30
5. ( m + 5)( m − 6)
6.
(x − 4)(x + 2)
7. (k + 5)(6k + 3)
8.
(
Section 7.3
1
2
)(
3
2
2u + — u − —
)
Multiplying Polynomials
343
EXAMPLE
4
Multiplying a Binomial and a Trinomial
Find (x + 5)(x 2 − 3x − 2).
x 2 − 3x − 2
×
x+5
Multiply 5(x 2 − 3x − 2).
Multiply x(x 2 − 3x − 2).
Align like terms vertically.
2
Distributive Property
2
x − 3x − 2x
Distributive Property
x 3 + 2x 2 − 17x − 10
Combine like terms.
5x − 15x − 10
3
The product is x 3 + 2x 2 − 17x − 10.
EXAMPLE
5
Real-Life Application
In hockey, a goalie behind the goal line can only play a puck in
a trapezoidal region.
a. Write a polynomial that represents the area of the
trapezoidal region.
1
2
1
2
Substitute.
1
2
Combine like terms.
—h(b1 + b2) = —(x − 7)[x + (x + 10)]
= —(x − 7)(2x + 10)
F
O
I
L
1
= — [2x 2 + 10x + (−14x) + (−70)] Use the FOIL Method.
2
x ft
1
2
(x − 7) ft
(x + 10) ft
= — (2x 2 − 4x − 70)
Combine like terms.
= x 2 − 2x − 35
Distributive Property
b. Find the area of the trapezoidal region when the shorter base
is 18 feet.
Find the value of x 2 − 2x − 35 when x = 18.
x 2 − 2x − 35 = 182 − 2(18) − 35
Substitute 18 for x.
= 324 − 36 − 35
Simplify.
= 253
Subtract.
The area of the trapezoidal region is 253 square feet.
Find the product.
Exercises 40–45
9. ( x + 1)( x 2 + 5x + 8)
10.
(n − 3)(n2 − 2n + 4)
11. WHAT IF? How does the polynomial in Example 5 change
if the longer base is extended by 1 foot? Explain.
344
Chapter 7
Polynomial Equations and Factoring
Exercises
7.3
Help with Homework
1. VOCABULARY Describe two ways to find the product of two binomials.
2. WRITING Explain how the letters of the word FOIL can help you remember
how to multiply two binomials.
6)=3
9+(- 3)=
3+(- 9)=
4+(- =
1)
9+(-
Write the product of the two binomials shown by the algebra tiles.
4. (−x + 3)(2x − 1) =
3. (x − 2)(x + 2) =
Use the Distributive Property to find the product.
1
5. (x + 1)(x + 3)
6. ( y + 6)( y + 4)
7. (z − 5)(z + 3)
8. (a + 8)(a − 3)
9. (g − 7)(g − 2)
10. (n − 6)(n − 4)
11. (3m + 1)(m + 9)
12. (2p − 4)(3p + 2)
14. ERROR ANALYSIS Describe and correct the
error in finding the product.
13. (6 − 5s)(2 − s)
✗
(t − 2)(t + 5) = t − 2(t + 5)
= t − 2t − 10
= −t − 10
15. CALCULATOR The width of a calculator
can be represented by (3x + 1) inches. The
length of the calculator is twice the width.
Write a polynomial that represents the area
of the calculator.
Section 7.3
Multiplying Polynomials
345
Use a table to find the product.
2 16. (x + 3)(x + 1)
19. (−3 + 2 j )(4 j − 7)
17. ( y + 10)( y − 5)
18. (h − 8)(h − 9)
20. (5c + 6)(6c + 5)
21. (5d − 12)(−7 + 3d)
Use the FOIL Method to find the product.
3 22. (b + 3)(b + 7)
23. (w + 9)(w + 6)
24. (k + 5)(k − 1)
25. (x − 4)(x + 8)
26. (q − 3)(q − 4)
27. (z − 5)(z − 9)
28. (t + 2)(2t + 1)
29. (5v − 3)(2v + 4)
30. (9 − r)(2 − 3r)
31. ERROR ANALYSIS Describe and correct the error in finding the product.
✗
(r + 6)(r − 7 ) = r (r) + r (7) + 6(r ) + 6(7)
= r 2 + 7r + 6r + 42
= r 2 + 13r + 42
32. OPEN-ENDED Write two binomials whose product includes the term 12.
(x + 10) yd
33. SOCCER The soccer field is rectangular.
a. Write a polynomial that represents
the area of the soccer field.
b. Use the polynomial in part (a) to find
the area of the field when x = 90.
(x − 30) yd
c. A groundskeeper mows 200
square yards in 3 minutes. How
long does it take the groundskeeper to mow the field?
Write a polynomial that represents the area of the shaded region.
34.
35.
36.
y−3
y−2
2p − 6
p+1
x−5
x−1
x−4
x+2
Find the product.
37. (n + 3)(2n 2 + 1)
4 40. (x − 4)(x 2 − 3x + 2)
43. (t 2 − 5t + 1)(−3 + t)
38. (x + y )(2x − y )
39. (2r + s)(r − 3s)
41. (f 2 + 4f − 8)(f − 1)
42. (3 + i )(i 2 + 8i − 2)
44. (b − 4)(5b 2 − 5b + 4)
45. (3e 2 − 5e + 7)(6e + 1)
46. REASONING Can you use the FOIL method to multiply a binomial by a trinomial?
a trinomial by a trinomial? Explain your reasoning.
346
Chapter 7
Polynomial Equations and Factoring
47. AMUSEMENT PARK You go to an amusement park
(x + 1) times each year and pay (x + 40) dollars each
time, where x is the number of years after 2011.
a. Write a polynomial that represents your yearly
admission cost.
b. What is your yearly admission cost in 2013?
48. PRECISION You use the Distributive Property to
multiply (x + 3)(x − 5). Your friend uses the
FOIL Method to multiply (x − 5)(x + 3). Should
your answers be equivalent? Justify your answer.
49. REASONING The product of (x + m)(x + n) is
x 2 + bx + c.
a. What do you know about m and n when c < 0?
b. What do you know about m and n when c > 0?
50. PICTURE You design the wooden picture frame and paint the front surface.
a. Write a polynomial that represents the area of wood you paint.
b. You design the picture frame to display a 5-inch by 8-inch
photograph. How much wood do you paint?
(x à 4) in.
x in.
(x à 3) in.
(x à 7) in.
51.
The shipping container is
a rectangular prism. Write a polynomial
that represents the volume of the container.
(x + 2) ft
(4x − 3) ft
(x + 1) ft
Write the polynomial in standard form. Identify the degree and classify the
polynomial by the number of terms. (Section 7.1)
52. 2x − 5x 2 − x 3
5
7
53. z 2 − — z
54. −15y 7
55. MULTIPLE CHOICE Which system of linear equations
does the graph represent? (Section 4.1)
A y = 3x + 4
○
y = −2x − 6
C y = −x + 7
○
y = 4x − 8
B y = 2x + 1
○
y
2
1
Ź4 Ź3 Ź2
1
2 x
y = −x − 2
Ź3
D y = x + 10
○
Ź4
y = −3x + 2
Section 7.3
Multiplying Polynomials
347