Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

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California StandardsCalifornia Standards

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Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Warm UpMultiply. Write each product as one power.

1. x · x2. 62 · 63

3. k2 · k8

4. 195 · 192

5. m · m5

6. 266 · 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

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Preview of Algebra 1 10.0 Students add, subtract,

multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.Also covered: 7AF1.2, 7AF1.3, 7AF2.2

California Standards

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

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

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 1: Multiplying Monomials

A. (2x3y2)(6x5y3)

(2x3y2)(6x5y3)

12x8y5Multiply coefficients. Addexponents that have the same base.

B. (9a5b7)(–2a4b3)

(9a5b7)(–2a4b3)

–18a9b10Multiply coefficients. Addexponents that have the same base.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 1

Multiply.

A. (5r4s3)(3r3s2)

(5r4s3)(3r3s2)

15r7s5Multiply coefficients. Addexponents that have the same base.

B. (7x3y5)(–3x3y2)

(7x3y5)(–3x3y2)

–21x6y7Multiply coefficients. Addexponents that have the same base.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

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

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 2: Multiplying a Polynomial by a Monomial

A. 3m(5m2 + 2m)

3m(5m2 + 2m)

15m3 + 6m2

Multiply each term in parentheses by 3m.

B. –6x2y3(5xy4 + 3x4)

–6x2y3(5xy4 + 3x4)

–30x3y7 – 18x6y3

Multiply each term in parentheses by –6x2y3.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 2: Multiplying a Polynomial by a Monomial

C. –5y3(y2 + 6y – 8)

–5y3(y2 + 6y – 8)

–5y5 – 30y4 + 40y3

Multiply each term in parentheses by –5y3.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

When multiplying a polynomial by a negative monomial, be sure to distribute the negative sign.

Helpful Hint

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 2

Multiply.A. 4r(8r3 + 16r)

4r(8r3 + 16r)

32r4 + 64r2

Multiply each term in parentheses by 4r.

B. –3a3b2(4ab3 + 4a2)

–3a3b2(4ab3 + 4a2)

–12a4b5 – 12a5b2

Multiply each term in parentheses by –3a3b2.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 2

Multiply.

C. –2x4(x3 + 4x + 3)

–2x4(x3 + 4x + 3)

–2x7 – 8x5 – 6x4

Multiply each term in parentheses by –2x4.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

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.

Additional Example 3: Problem Solving Application

11 Understand the Problem

If the width of the frame is w and the length is 3w – 8, then the area is w(3w – 8) or length times width. The answer will be a value of w that makes the area of the frame equal to 60 in2.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Additional Example 3 Continued

22 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.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Additional Example 3 Continued

Solve33

w(3w – 8) = 3w2 – 8w Distributive Property

w 3 4 5 6

3w2 – 8w 3(32) – 8(3)= 3

3(42) – 8(4)= 16

3(52) – 8(5)= 35

3(62) – 8(6)= 60

The width should be 6 in. and the length should be 10 in.

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Look Back44

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.

Additional Example 3 Continued

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

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.

11 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.

12

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 3 Continued

22 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.

12

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 3 Continued

Solve33

b 9 10 11 12

92 = 81 102 = 100 112 = 121

The length of the base should be 12 in.

b(2b) = b212

b2 122 = 144

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Check It Out! Example 3 Continued

Look Back44

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.

12

Holt CA Course 1

12-5 Multiplying Polynomials by Monomials

Lesson QuizMultiply.

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.

–20x3y5z3

6a3b3

6n4 – 9n2

–15p2q + 30p3

l = 7 ft, w = 9 ft

–4x3y – 4xy3 + 4xy