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6-2 6-2 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra Holt Algebra 22 6-2 Multiplying Polynomials Warm Up Multiply. 1. x(x3) x4 2. 3x2(x5) 3x7 3. 2(5x3) 10x3 4. x(6x2) 6x3 5. xy(7x2) 7x3y 6. 3y2(–3y) –9y3 Holt Algebra 2 6-2 Multiplying Polynomials Holt Algebra 2 6-2 Multiplying Polynomials Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers. Holt Algebra 2 6-2 Multiplying Polynomials Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2 + 3) 4y2(y2 + 3) 4y2 y2 + 4y2 3 4y4 + 12y2 Distribute. Multiply. B. fg(f4 + 2f3g – 3f2g2 + fg3) fg(f4 + 2f3g – 3f2g2 + fg3) fg f4 + fg 2f3g – fg 3f2g2 + fg fg3 Distribute. f5g + 2f4g2 – 3f3g3 + f2g4 Multiply. Holt Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 1 Find each product. a. 3cd2(4c2d – 6cd + 14cd2) 3cd2(4c2d – 6cd + 14cd2) 3cd2 4c2d – 3cd2 6cd + 3cd2 14cd2 12c3d3 – 18c2d3 + 42c2d4 Distribute. Multiply. b. x2y(6y3 + y2 – 28y + 30) x2y(6y3 + y2 – 28y + 30) x2y 6y3 + x2y y2 – x2y 28y + x2y 30 Distribute. 6x2y4 + x2y3 – 28x2y2 + 30x2y Multiply. Holt Algebra 2 6-2 Multiplying Polynomials To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first. Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified. Holt Algebra 2 6-2 Multiplying Polynomials Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) a3 – 8a2 + 17a – 6 Holt Algebra 2 Combine like terms. 6-2 Multiplying Polynomials Example 2B: Multiplying Polynomials Find the product. (y2 – 7y + 5)(y2 – y – 3) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y2 –y –3 The top left corner is the first 2 4 3 2 y y –y –3y term in the product. Combine terms along diagonals to get –7y –7y3 7y2 21y the middle terms. The bottom right corner is the last term in 5 2 5y –5y –15 the product. y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15 y4 – 8y3 + 9y2 + 16y – 15 Holt Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 2a Find the product. (3b – 2c)(3b2 – bc – 2c2) 9b3 – 9b2c – 4bc2 + 4c3 Holt Algebra 2 Combine like terms. 6-2 Multiplying Polynomials Check It Out! Example 2b Find the product. (x2 – 4x + 1)(x2 + 5x – 2) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. x2 –4x 1 The top left corner is the first 2 4 3 2 x x –4x x term in the product. Combine terms along diagonals to get 2 5x 5x3 –20x 5x the middle terms. The bottom right corner is the last term in –2 –2x2 8x –2 the product. x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2 x4 + x3 – 21x2 + 13x – 2 Holt Algebra 2 6-2 Multiplying Polynomials Example 3: Business Application A standard Burly Box is p ft by 3p ft by 4p ft. A large Burly Box has 1.5 ft added to each dimension. Write a polynomial V(p) in standard form that can be used to find the volume of a large Burly Box. The volume of a large Burly Box is the product of the area of the base and height. V(p) = A(p) h(p) The area of the base of the large Burly Box is the product of the length and width of the box. A(p) = l(p) w(p) The length, width, and height of the large Burly Box are greater than that of the standard Burly Box. l(p) = p + 1.5, w(p) = 3p + 1.5, h(p) = 4p + 1.5 Holt Algebra 2 6-2 Multiplying Polynomials Example 3: Business Application Solve A(p) = l(p) w(p). p + 1.5 3p + 1.5 1.5p + 2.25 3p2 + 4.5p 3p2 + 6p + 2.25 Solve V(p) = A(p) h(p). 3p2 + 6p + 2.25 4p + 1.5 4.5p2 + 9p + 3.375 12p3 + 24p2 + 9p 12p3 + 28.5p2 + 18p + 3.375 The volume of a large Burly Box can be modeled by V(p) = 12p3 + 28.5p2 + 18p + 3.375 Holt Algebra 2 6-2 Multiplying Polynomials Notice the coefficients of the variables in the final product of (a + b)3. these coefficients are the numbers from the third row of Pascal's triangle. Each row of Pascal’s triangle gives the coefficients of the corresponding binomial expansion. The pattern in the table can be extended to apply to the expansion of any binomial of the form (a + b)n, where n is a whole number. Holt Algebra 2 6-2 Multiplying Polynomials This information is formalized by the Binomial Theorem, which you will study further in Chapter 11. Holt Algebra 2 6-2 Multiplying Polynomials Lesson Quiz Find each product. 1. 5jk(k – 2j) 5jk2 – 10j2k 2. (2a3 – a + 3)(a2 + 3a – 5) 2a5 + 6a4 – 11a3 + 14a – 15 3. The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost. –0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10 4. Find the product. (y – 5)4 y4 – 20y3 + 150y2 – 500y + 625 5. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3 Holt Algebra 2