Multiplying Polynomials 14 5 Multiplying Polynomials by Monomials

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

14 -5 Multiplying Polynomials by Monomials Warm Up Multiply. Write each product as one power. 1. x · x x 2 2. 62 · 63 65 3. k 2 · k 8 k 10 4. 195 · 192 197 5. m · m 5 m 6 6. 266 · 265 2611 7. Find the volume of a rectangular prism that measures 5 cm by 2 cm by 6 cm. 60 cm 3 Course 3

14 -5 Multiplying Polynomials by Monomials 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 Course 3

14 -5 Multiplying Polynomials by Monomials Learn to multiply polynomials by monomials. Course 3

14 -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. (5 m 2 n 3)(6 m 3 n 6) = 5 · 6 · m 2 + 3 n 3 + 6 = 30 m 5 n 9 Course 3

14 -5 Multiplying Polynomials by Monomials Additional Example 1: Multiplying Monomials Multiply. A. (2 x 3 y 2)(6 x 5 y 3) 12 x 8 y 5 Multiply coefficients and add exponents. B. (9 a 5 b 7)(– 2 a 4 b 3) – 18 a 9 b 10 Course 3 Multiply coefficients and add exponents.

14 -5 Multiplying Polynomials by Monomials Check It Out: Example 1 Multiply. A. (5 r 4 s 3)(3 r 3 s 2) 15 r 7 s 5 Multiply coefficients and add exponents. B. (7 x 3 y 5)(– 3 x 3 y 2) – 21 x 6 y 7 Course 3 Multiply coefficients and add exponents.

14 -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. Course 3

14 -5 Multiplying Polynomials by Monomials Additional Example 2: Multiplying a Polynomial by a Monomial Multiply. A. 3 m(5 m 2 + 2 m) 15 m 3 + 6 m 2 Multiply each term in parentheses by 3 m. B. – 6 x 2 y 3(5 xy 4 + 3 x 4) – 30 x 3 y 7 – 18 x 6 y 3 Course 3 Multiply each term in parentheses by – 6 x 2 y 3.

14 -5 Multiplying Polynomials by Monomials Additional Example 2: Multiplying a Polynomial by a Monomial Multiply. C. – 5 y 3(y 2 + 6 y – 8) – 5 y 5 – 30 y 4 + 40 y 3 Course 3 Multiply each term in parentheses by – 5 y 3.

14 -5 Multiplying Polynomials by Monomials Check It Out: Example 2 Multiply. A. 4 r(8 r 3 + 16 r) 32 r 4 + 64 r 2 Multiply each term in parentheses by 4 r. B. – 3 a 3 b 2(4 ab 3 + 4 a 2) – 12 a 4 b 5 – 12 a 5 b 2 Course 3 Multiply each term in parentheses by – 3 a 3 b 2.

14 -5 Multiplying Polynomials by Monomials Check It Out: Example 2 Multiply. C. – 2 x 4(x 3 + 4 x + 3) – 2 x 7 – 8 x 5 – 6 x 4 Course 3 Multiply each term in parentheses by – 2 x 4.

14 -5 Multiplying Polynomials by Monomials 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 in 2. 1 Understand the Problem If the width of the frame is w and the length is 3 w – 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 in 2. Course 3

14 -5 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(3 w – 8) another way. Use substitution to complete the table. Course 3

14 -5 Multiplying Polynomials by Monomials Additional Example 3 Continued 3 Solve w(3 w – 8) = 3 w 2 – 8 w Distributive Property w 3 w 2 3 – 8 w 3(32) =3 – 8(3) 4 3(42) = 16 – 8(4) 5 3(52) = 35 6 – 8(5) 3(62) – 8(6) = 60 The width should be 6 in. and the length should be 10 in. Course 3

14 -5 Multiplying Polynomials by Monomials Additional Example 3 Continued 4 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 in 2. The answer is reasonable. Course 3

14 -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 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 = 2 b. Thus the area would be 12 b(2 b). The answer will be a value of b that makes the area equal to 144 in 2. Course 3

14 -5 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 1 expression 2 b(2 b) another way. Use substitution to complete the table. Course 3

14 -5 Multiplying Polynomials by Monomials Check It Out: Example 3 Continued 3 Solve 1 b(2 b) 2 b b 2 = b 2 9 10 11 92 = 81 102 = 100 112 = 121 122 = 144 The length of the base should be 12 in. Course 3 12

14 -5 Multiplying Polynomials by Monomials Check It Out: Example 3 Continued 4 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 in 2. The answer is reasonable. Course 3

14 -5 Multiplying Polynomials by Monomials Multiply. Lesson Quiz 1. (3 a 2 b)(2 ab 2) 6 a 3 b 3 2. (4 x 2 y 2 z)(– 5 xy 3 z 2) – 20 x 3 y 5 z 3 3. 3 n(2 n 3 – 3 n) 6 n 4 – 9 n 2 4. – 5 p 2(3 q – 6 p) – 15 p 2 q + 30 p 3 5. – 2 xy(2 x 2 + 2 y 2 – 2) – 4 x 3 y – 4 xy 3 + 4 xy 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. l = 7 ft, w = 9 ft Course 3