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Multiply Polynomials by Monomials: Lesson Presentation

Learn how to multiply polynomials by monomials by multiplying coefficients and adding exponents. Use the Distributive Property to multiply a polynomial by a monomial.

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Multiply Polynomials by Monomials: Lesson Presentation

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  1. Multiplying Polynomials by Monomials 14-5 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 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 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 Multiply. A. (2x3y2)(6x5y3) (2x3y2)(6x5y3) Multiply coefficients and add exponents. 12x8y5 B. (9a5b7)(–2a4b3) (9a5b7)(–2a4b3) Multiply coefficients and add exponents. –18a9b10

  7. Check It Out: Example 1 Multiply. A. (5r4s3)(3r3s2) (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. Multiply every term of the polynomial by the monomial.

  9. Additional Example 2: Multiplying a Polynomial by a Monomial Multiply. A. 3m(5m2 + 2m) Multiply each term in parentheses by 3m. 3m(5m2+ 2m) 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 Multiply. C. –5y3(y2 + 6y– 8) –5y3(y2 + 6y – 8) Multiply each term in parentheses by –5y3. –5y5– 30y4 + 40y3

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

  12. 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. 1 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. 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. Make a Plan 2 Additional Example 3 Continued 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. 3 Solve Additional Example 3 Continued w(3w– 8) = 3w2– 8w Distributive Property 5 4 6 w 3 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. 4 Additional Example 3 Continued 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. 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 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.

  18. 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 2 Check It Out: Example 3 Continued

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

  20. 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 4 Check It Out: Example 3 Continued Look Back

  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

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