13-5 Multiplying Polynomials by Monomials

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Pre-Algebra

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

13-5 Multiplying Polynomials by Monomials

197

m6

2611

60 cm3

Pre-Algebra

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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Learn to multiply polynomials by monomials.

Pre-Algebra

13-5 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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 1: Multiplying Monomials

A. (2x3y2)(6x5y3)

(2x3y2)(6x5y3)

12x8y5

Multiply coefficients and addexponents.

B. (9a5b7)(–2a4b3)

(9a5b7)(–2a4b3)

–18a9b10

Multiply coefficients and addexponents.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 1

Insert Lesson Title Here

Multiply.

A. (5r4s3)(3r3s2)

(5r4s3)(3r3s2)

15r7s5

Multiply coefficients and addexponents.

B. (7x3y5)(–3x3y2)

(7x3y5)(–3x3y2)

–21x6y7

Multiply coefficients and addexponents.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 2A & 2B: 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.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Multiply.

Additional Example 2C: 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.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 2A & 2B

Multiply.

Insert Lesson Title Here

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.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 2C

Insert Lesson Title Here

Multiply.

C. –2x4(x3 + 4x + 3)

–2x4(x3 + 4x + 3)

–2x7 – 8x5 – 6x4

Multiply each term in parentheses by –2x4.

Pre-Algebra

13-5 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(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.

Pre-Algebra

13-5 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.

Pre-Algebra

13-5 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.

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Look Back44

If the width is 6 inches and the length is 3 times that minus 8 or 10 inches, then the area would be 6 · 10 = 60 in2. The answer is reasonable.

Pre-Algebra

Additional Example 3 Continued

13-5 Multiplying Polynomials by Monomials

Try This: Example 3

Insert Lesson Title Here

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 base b is equal to 2 times height, 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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 3 Continued

Insert Lesson Title Here

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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 3 Continued

Insert Lesson Title Here

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

Pre-Algebra

13-5 Multiplying Polynomials by Monomials

Try This: Example 3 Continued

Insert Lesson Title Here

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

Pre-Algebra

13-5 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

Insert Lesson Title Here

6n4 – 9n2

–15p2q + 30p3

Pre-Algebra

l = 7 ft, w = 9 ft

–4x3y – 4xy3 + 4xy

13-5 Multiplying Polynomials by Monomials