1. Multiplying Polynomials by Monomials
14-5
Multiplying Polynomials by Monomials
Warm Up
Problem of the Day
Lesson Presentation
Course 3

2. Warm Up
Multiply. Write each product as one power.
1. x · x
2. 62 · 63
3. k2 · k8
· 192
5. m · m5
· 265
7. Find the volume of a rectangular prism that measures 5 cm by 2 cm by 6 cm. x2 65
k10
197 m6
2611
60 cm3
Course 3

3. Problem of the Day
Charlie added 3 binomials, 2 trinomials, and 1 monomial. What is the greatest possible number of terms in the sum? 13

4. Learn to multiply polynomials by monomials.

5. Remember that when you multiply two powers with the same bases, you add the exponents. To multiply two monomials, multiply the coefficients and add the exponents of the variables that are the same.
(5m2n3)(6m3n6) = 5 · 6 · m2 + 3n3 + 6 = 30m5n9

6. Additional Example 1: Multiplying Monomials
A. (2x3y2)(6x5y3)
(2x3y2)(6x5y3)
Multiply coefficients and add
exponents.
12x8y5
B. (9a5b7)(–2a4b3)
(9a5b7)(–2a4b3)
Multiply coefficients and add
exponents.
–18a9b10

7. –21x6y7 Check It Out: Example 1 Multiply. A. (5r4s3)(3r3s2)
Multiply coefficients and add
exponents.
15r7s5
B. (7x3y5)(–3x3y2)
(7x3y5)(–3x3y2)
Multiply coefficients and add
exponents.
–21x6y7

8. To multiply a polynomial by a monomial, use the Distributive Property
To multiply a polynomial by a monomial, use the Distributive Property. Multiply every term of the polynomial by the monomial.

9. Additional Example 2: Multiplying a Polynomial by a Monomial
A. 3m(5m2 + 2m)
3m(5m2 + 2m)
Multiply each term in
parentheses by 3m.
15m3 + 6m2
B. –6x2y3(5xy4 + 3x4)
–6x2y3(5xy4 + 3x4)
Multiply each term in
parentheses by –6x2y3.
–30x3y7 – 18x6y3

10. Additional Example 2: Multiplying a Polynomial by a Monomial
C. –5y3(y2 + 6y – 8)
–5y3(y2 + 6y – 8)
Multiply each term in
parentheses by –5y3.
–5y5 – 30y4 + 40y3

11. –3a3b2(4ab3 + 4a2) –12a4b5 – 12a5b2 Check It Out: Example 2 Multiply.
A. 4r(8r3 + 16r)
4r(8r3 + 16r)
Multiply each term in
parentheses by 4r.
32r4 + 64r2
B. –3a3b2(4ab3 + 4a2)
–3a3b2(4ab3 + 4a2)
Multiply each term in
parentheses by –3a3b2.
–12a4b5 – 12a5b2

12. –2x4(x3 + 4x + 3) –2x7 – 8x5 – 6x4 Check It Out: Example 2 Multiply.
C. –2x4(x3 + 4x + 3)
–2x4(x3 + 4x + 3)
Multiply each term in
parentheses by –2x4.
–2x7 – 8x5 – 6x4

13. Understand the Problem
Additional Example 3: Problem Solving Application
The length of a picture in a frame is 8 in. less than three times its width. Find the length and width if the area is 60 in2. 1
Understand the Problem
If the width of the frame is w and the length is 3w – 8, then the area is w(w – 8) or length times width. The answer will be a value of w that makes the area of the frame equal to 60 in2.

14. Additional Example 3 Continued2
Make a Plan
You can make a table of values for the polynomial to try to find the value of a w. Use the Distributive Property to write the expression w(3w – 8) another way. Use substitution to complete the table.

15. Additional Example 3 Continued
Solve 3
w(3w – 8) = 3w2 – 8w
Distributive Property w 3 4 5 6
3(32) – 8(3)
= 3
3(42) – 8(4)
= 16
3(52) – 8(5)
= 35
3(62) – 8(6)
= 60
3w2 – 8w
The width should be 6 in. and the length should be 10 in.

16. Additional Example 3 Continued4
Look Back
If the width is 6 inches and the length is 3 times the width minus 8, or 10 inches, then the area would be 6 · 10 = 60 in2. The answer is reasonable.

17. Understand the Problem
Check It Out: Example 3
The height of a triangle is twice its base. Find the base and the height if the area is 144 in2. 1
Understand the Problem
The formula for the area of a triangle is one-half base times height. Since the height h is equal to 2 times base, h = 2b. Thus the area would be b(2b). The answer will be a value of b that makes the area equal to 144 in2. 1 2

18. Check It Out: Example 3 Continued2
Make a Plan
You can make a table of values for the polynomial to find the value of b. Write the expression b(2b) another way. Use substitution to complete the table. 1 2

19. Check It Out: Example 3 Continued
Solve 3
b(2b) = b2 1 2 b 9 10 11 12 b2
92 = 81
102 = 100
112 = 121
122 = 144
The length of the base should be 12 in.

20. Check It Out: Example 3 Continued4
Look Back
If the height is twice the base, and the base is 12 in., the height would be 24 in. The area would be · 12 · 24 = 144 in2. The answer is reasonable. 1 2

21. Lesson Quiz
Multiply.
1. (3a2b)(2ab2)
2. (4x2y2z)(–5xy3z2)
3. 3n(2n3 – 3n)
4. –5p2(3q – 6p)
5. –2xy(2x2 + 2y2 – 2)
6. The width of a garden is 5 feet less than 2 times its length. Find the garden’s length and width if its area is 63 ft2.
6a3b3
–20x3y5z3
6n4 – 9n2
–15p2q + 30p3
–4x3y – 4xy3 + 4xy
l = 7 ft, w = 9 ft