13 5 Multiplying Polynomials by Monomials Warm Up

13 -5 Multiplying Polynomials by Monomials Warm Up Problem of the Day Lesson Presentation Pre-Algebra

13 -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 Pre-Algebra

13 -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 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. (5 m 2 n 3)(6 m 3 n 6) = 5 · 6 · m 2+3 n 3+6 = 30 m 5 n 9 Pre-Algebra

13 -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 Pre-Algebra Multiply coefficients and add exponents.

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: 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 Pre-Algebra Multiply coefficients and add exponents.

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 Additional Example 2 A & 2 B: 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 Pre-Algebra Multiply each term in parentheses by – 6 x 2 y 3.

13 -5 Multiplying Polynomials by Monomials Additional Example 2 C: 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 Pre-Algebra Multiply each term in parentheses by – 5 y 3.

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: Example 2 A & 2 B 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 Pre-Algebra Multiply each term in parentheses by – 3 a 3 b 2.

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: Example 2 C Multiply. C. – 2 x 4(x 3 + 4 x + 3) – 2 x 7 – 8 x 5 – 6 x 4 Pre-Algebra Multiply each term in parentheses by – 2 x 4.

13 -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. Pre-Algebra

13 -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. Pre-Algebra

13 -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 – 8 w 3 3(32) =3 – 8(3) 4 3(42) = 16 – 8(4) 5 3(52) = 35 – 8(5) 3(62) – 8(6) The width should be 6 in. and the length should be 10 in. Pre-Algebra 6 = 60

13 -5 Multiplying Polynomials by Monomials Additional Example 3 Continued 4 Look Back 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 in 2. The answer is reasonable. Pre-Algebra

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: 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 base b is equal to 2 times height, 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. Pre-Algebra

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: 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. Pre-Algebra

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: 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. Pre-Algebra 12

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: 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. Pre-Algebra

13 -5 Multiplying Polynomials by Monomials Insert Lesson Title Here 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 Pre-Algebra