Holt CA Course Multiplying Polynomials by Monomials Warm Up Warm Up California Standards California Standards Lesson Presentation Lesson PresentationPreview

Holt CA Course Multiplying Polynomials by Monomials Warm Up Multiply. Write each product as one power. 1. x · x · k 2 · k · m · m · Find the volume of a rectangular prism that measures 5 cm by 2 cm by 6 cm. x2x k m6m cm 3

Holt CA Course Multiplying Polynomials by Monomials Preview of Algebra 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 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. (5m 2 n 3 )(6m 3 n 6 ) = 5 · 6 · m n = 30m 5 n 9

Holt CA Course Multiplying Polynomials by Monomials Multiply. Additional Example 1: Multiplying Monomials A. (2x 3 y 2 )(6x 5 y 3 ) (2x 3 y 2 )(6x 5 y 3 ) 12x 8 y 5 Multiply coefficients. Add exponents that have the same base. B. (9a 5 b 7 )( – 2a 4 b 3 ) (9a 5 b 7 )( – 2a 4 b 3 ) – 18a 9 b 10 Multiply coefficients. Add exponents that have the same base.

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 1 Multiply. A. (5r 4 s 3 )(3r 3 s 2 ) (5r 4 s 3 )(3r 3 s 2 ) 15r 7 s 5 Multiply coefficients. Add exponents that have the same base. B. (7x 3 y 5 )( – 3x 3 y 2 ) (7x 3 y 5 )( – 3x 3 y 2 ) – 21x 6 y 7 Multiply coefficients. Add exponents that have the same base.

Holt CA Course 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 Multiplying Polynomials by Monomials Multiply. Additional Example 2: Multiplying a Polynomial by a Monomial A. 3m(5m 2 + 2m) 3m(5m 2 + 2m) 15m 3 + 6m 2 Multiply each term in parentheses by 3m. B. – 6x 2 y 3 (5xy 4 + 3x 4 ) – 6x 2 y 3 (5xy 4 + 3x 4 ) – 30x 3 y 7 – 18x 6 y 3 Multiply each term in parentheses by – 6x 2 y 3.

Holt CA Course Multiplying Polynomials by Monomials Multiply. Additional Example 2: Multiplying a Polynomial by a Monomial C. – 5y 3 (y 2 + 6y – 8) – 5y 3 (y 2 + 6y – 8) – 5y 5 – 30y y 3 Multiply each term in parentheses by – 5y 3.

Holt CA Course Multiplying Polynomials by Monomials When multiplying a polynomial by a negative monomial, be sure to distribute the negative sign. Helpful Hint

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 2 Multiply. A. 4r(8r r) 4r(8r r) 32r r 2 Multiply each term in parentheses by 4r. B. – 3a 3 b 2 (4ab 3 + 4a 2 ) – 3a 3 b 2 (4ab 3 + 4a 2 ) – 12a 4 b 5 – 12a 5 b 2 Multiply each term in parentheses by – 3a 3 b 2.

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 2 Multiply. C. – 2x 4 (x 3 + 4x + 3) – 2x 4 (x 3 + 4x + 3) – 2x 7 – 8x 5 – 6x 4 Multiply each term in parentheses by – 2x 4.

Holt CA Course 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 in 2. Additional Example 3: Problem Solving Application 1 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 in 2.

Holt CA Course Multiplying Polynomials by Monomials Additional Example 3 Continued 2 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 Multiplying Polynomials by Monomials Additional Example 3 Continued Solve 3 w(3w – 8) = 3w 2 – 8w Distributive Property w w 2 – 8w 3(3 2 ) – 8(3) = 3 3(4 2 ) – 8(4) = 16 3(5 2 ) – 8(5) = 35 3(6 2) – 8(6) = 60 The width should be 6 in. and the length should be 10 in.

Holt CA Course Multiplying Polynomials by Monomials Look Back 4 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 in 2. The answer is reasonable. Additional Example 3 Continued

Holt CA Course 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 in 2. 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 in

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 3 Continued 2 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. 1212

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 3 Continued Solve 3 b = = = 121 The length of the base should be 12 in. b(2b) = b b2b = 144

Holt CA Course Multiplying Polynomials by Monomials Check It Out! Example 3 Continued Look Back 4 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 in 2. The answer is reasonable. 1212

Holt CA Course Multiplying Polynomials by Monomials Lesson Quiz Multiply. 1. (3a 2 b)(2ab 2 ) 2. (4x 2 y 2 z)(–5xy 3 z 2 ) 3. 3n(2n 3 – 3n) 4. –5p 2 (3q – 6p) 5. –2xy(2x 2 + 2y 2 – 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 ft 2. –20x 3 y 5 z 3 6a3b36a3b3 6n 4 – 9n 2 –15p 2 q + 30p 3 l = 7 ft, w = 9 ft –4x 3 y – 4xy 3 + 4xy