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6-2 6-2 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra Holt McDougal Algebra 22 6-2 Multiplying Polynomials Warm Up Multiply. 1. x(x3) 2. 3x2(x5) 3. 2(5x3) 4. x(6x2) 5. xy(7x2) 6. 3y2(–3y) Holt McDougal Algebra 2 6-2 Multiplying Polynomials Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers. Holt McDougal Algebra 2 6-2 Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents. Holt McDougal Algebra 2 6-2 Multiplying Polynomials Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2 + 3) B. fg(f4 + 2f3g – 3f2g2 + fg3) Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 1 Find each product. a. 3cd2(4c2d – 6cd + 14cd2) b. x2y(6y3 + y2 – 28y + 30) Holt McDougal 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 McDougal Algebra 2 6-2 Multiplying Polynomials Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 1 Multiply horizontally. Holt McDougal Algebra 2 6-2 Multiplying Polynomials Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 2 Multiply vertically. Holt McDougal Algebra 2 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 McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 2a Find the product. (3b – 2c)(3b2 – bc – 2c2) Holt McDougal Algebra 2 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 McDougal 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. Holt McDougal Algebra 2 6-2 Multiplying Polynomials Example 3: Business Application Solve A(p) = l(p) w(p). Holt McDougal Algebra 2 Solve V(p) = A(p) h(p). 6-2 Multiplying Polynomials Check It Out! Example 3 Mr. Silva manages a manufacturing plant. From 1990 through 2005 the number of units produced (in thousands) can be modeled by N(x) = 0.02x2 + 0.2x + 3. The average cost per unit (in dollars) can be modeled by C(x) = –0.004x2 – 0.1x + 3. Write a polynomial T(x) that can be used to model the total costs. Total cost is the product of the number of units and the cost per unit. T(x) = N(x) C(x) Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 3 Multiply the two polynomials. Holt McDougal Algebra 2 6-2 Multiplying Polynomials Example 4: Expanding a Power of a Binomial Find the product. (a + 2b)3 Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 4a Find the product. (x + 4)4 Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 4b Find the product. (2x – 1)3 Holt McDougal 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 McDougal Algebra 2 6-2 Multiplying Polynomials This information is formalized by the Binomial Theorem, which you will study further in Chapter 11. Holt McDougal Algebra 2 6-2 Multiplying Polynomials Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5)3 B. (6m – 8)3 Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 5 Expand each expression. a. (x + 2)3 b. (x – 4)5 Holt McDougal Algebra 2 6-2 Multiplying Polynomials Check It Out! Example 5 Expand the expression. c. (3x + 1)4 Holt McDougal Algebra 2 6-2 Multiplying Polynomials Lesson Quiz Find each product. 1. 5jk(k – 2j) 2. (2a3 – a + 3)(a2 + 3a – 5) 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. 4. Find the product. (y – 5)4 5. Expand the expression. (3a – b)3 Holt McDougal Algebra 2