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

Multiplying Polynomials by Monomials 13-5 Multiplying Polynomials by Monomials 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 Pre-Algebra

Multiplying Polynomials by Monomials 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

Learn to multiply polynomials by monomials. 13-5 Multiplying Polynomials by Monomials Learn to multiply polynomials by monomials. Pre-Algebra

Multiplying Polynomials by Monomials 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

Additional Example 1: Multiplying Monomials 13-5 Multiplying Polynomials by Monomials 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 Pre-Algebra

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

Multiplying Polynomials by Monomials 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

Additional Example 2A & 2B: Multiplying a Polynomial by a Monomial 13-5 Multiplying Polynomials by Monomials Additional Example 2A & 2B: Multiplying a Polynomial by a Monomial Multiply. A. 3m(5m2 + 2m) 3m(5m2 + 2m) Multiply each term in parentheses by 3m. 15m3 + 6m2 B. –6x2y3(5xy4 + 3x4) –6x2y3(5xy4 + 3x4) Multiply each term in parentheses by –6x2y3. –30x3y7 – 18x6y3 Pre-Algebra

Additional Example 2C: Multiplying a Polynomial by a Monomial 13-5 Multiplying Polynomials by Monomials Additional Example 2C: 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 Pre-Algebra

Insert Lesson Title Here 13-5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: Example 2A & 2B Multiply. A. 4r(8r3 + 16r) 4r(8r3 + 16r) Multiply each term in parentheses by 4r. 32r4 + 64r2 B. –3a3b2(4ab3 + 4a2) –3a3b2(4ab3 + 4a2) Multiply each term in parentheses by –3a3b2. –12a4b5 – 12a5b2 Pre-Algebra

Insert Lesson Title Here 13-5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: Example 2C Multiply. C. –2x4(x3 + 4x + 3) –2x4(x3 + 4x + 3) Multiply each term in parentheses by –2x4. –2x7 – 8x5 – 6x4 Pre-Algebra

Understand the Problem 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 in2. 1 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

Additional Example 3 Continued 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(3w – 8) another way. Use substitution to complete the table. Pre-Algebra

Additional Example 3 Continued 13-5 Multiplying Polynomials by Monomials Additional Example 3 Continued Solve 3 w(3w – 8) = 3w2 – 8w Distributive Property w 3 4 5 6 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. Pre-Algebra

Additional Example 3 Continued 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 in2. The answer is reasonable. Pre-Algebra

Understand the Problem 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 in2. 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 =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 Pre-Algebra

Insert Lesson Title Here 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 expression b(2b) another way. Use substitution to complete the table. 1 2 Pre-Algebra

Insert Lesson Title Here 13-5 Multiplying Polynomials by Monomials Insert Lesson Title Here Try This: Example 3 Continued Solve 3 b(2b) = b2 1 2 b 9 10 11 12 b2 92 = 81 102 = 100 112 = 121 122 = 144 The length of the base should be 12 in. Pre-Algebra

Insert Lesson Title Here 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 in2. The answer is reasonable. 1 2 Pre-Algebra

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