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Warm Up Multiply.

Warm Up Multiply. x 4. 1. x ( x 3 ). 3 x 7. 2. 3 x 2 ( x 5 ). 3. 2(5 x 3 ). 10 x 3. 4. x (6 x 2 ). 6 x 3. 5. xy (7 x 2 ). 7 x 3 y. 6. 3 y 2 (–3 y ). –9 y 3. Objectives. Multiply polynomials.

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Warm Up Multiply.

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  1. Warm Up Multiply. x4 1. x(x3) 3x7 2. 3x2(x5) 3. 2(5x3) 10x3 4. x(6x2) 6x3 5. xy(7x2) 7x3y 6. 3y2(–3y) –9y3

  2. Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

  3. To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

  4. Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4y2(y2+ 3) 4y2(y2+ 3) 4y2  y2+ 4y2 3 Distribute. 4y4 + 12y2 Multiply. B. fg(f4 + 2f3g – 3f2g2 + fg3) fg(f4 + 2f3g – 3f2g2 + fg3) Distribute. fgf4 + fg 2f3g – fg  3f2g2 + fgfg3 f5g + 2f4g2 – 3f3g3 + f2g4 Multiply.

  5. Check It Out! Example 1 Find each product. a. 3cd2(4c2d– 6cd + 14cd2) 3cd2(4c2d– 6cd + 14cd2) 3cd2  4c2d– 3cd2 6cd + 3cd2 14cd2 Distribute. 12c3d3 – 18c2d3 + 42c2d4 Multiply. b. x2y(6y3 + y2 – 28y + 30) x2y(6y3 + y2 – 28y + 30) Distribute. x2y 6y3 + x2yy2 – x2y  28y + x2y 30 6x2y4 + x2y3 – 28x2y2 + 30x2y Multiply.

  6. 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.

  7. Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 1 Multiply horizontally. (a – 3)(a2 – 5a + 2) Write polynomials in standard form. Distribute a and then –3. a(a2) + a(–5a)+ a(2) – 3(a2) – 3(–5a) –3(2) a3 – 5a2+ 2a – 3a2 + 15a – 6 Multiply. Add exponents. a3 – 8a2+ 17a – 6 Combine like terms.

  8. a2 – 5a + 2 a – 3 Example 2A: Multiplying Polynomials Find the product. (a – 3)(2 – 5a + a2) Method 2 Multiply vertically. Write each polynomial in standard form. – 3a2 + 15a – 6 Multiply (a2 – 5a + 2) by –3. a3 – 5a2+ 2a Multiply (a2 – 5a + 2) by a, and align like terms. a3 – 8a2+ 17a – 6 Combine like terms.

  9. y2 –y–3 y2 –7y 5 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. The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product. y4+ (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15 y4 – 8y3 + 9y2 + 16y – 15

  10. Check It Out! Example 2a Find the product. (3b – 2c)(3b2 – bc – 2c2) Multiply horizontally. Write polynomials in standard form. (3b – 2c)(3b2 – 2c2 – bc) Distribute 3b and then –2c. 3b(3b2) + 3b(–2c2)+ 3b(–bc) – 2c(3b2) – 2c(–2c2) – 2c(–bc) Multiply. Add exponents. 9b3 – 6bc2– 3b2c – 6b2c + 4c3+ 2bc2 9b3 – 9b2c – 4bc2 + 4c3 Combine like terms.

  11. x2 –4x1 x2 5x –2 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. The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product. x4+ (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2 x4 + x3 – 21x2 + 13x – 2

  12. Lesson Quiz Find each product. 1. 5jk(k – 2j) 2. (2a3– a + 3)(a2+ 3a – 5) 5jk2– 10j2k 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

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