Multiplying Polynomials 6-2 Multiplying Polynomials Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

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

Objectives Multiply polynomials.

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

Zero exponent, the answer is one and the base goes away. Exponent Rules When you multiply like bases, you add the exponents and the base stays the same. When you divide like bases, you subtract the exponents and the base stays the same. Power raised to a power, you multiply the powers and the base stays the same. Zero exponent, the answer is one and the base goes away. Negative exponent, send the base to the other side, and the exponent becomes positive.

Example 1: Multiplying a Monomial and a Polynomial Find each product. 4x(y - 3x) 4xy –12x2 y -3x 4x Distribute. 4xy –12x2 Multiply. B. fg(f4 + 2f3g – 3f2g2 ) f 4 2f3g – 3f2g2 f 5g 2f 4g2 – 3f 3g3 fg Distribute. Multiply. f 5g + 2f 4g2 – 3f 3g3

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) x2y  6y3 + x2y  y2 – x2y  28y + x2y  30 Distribute. 6x2y4 + x2y3 – 28x2y2 + 30x2y Multiply.

Example 2B: Multiplying Polynomials Find the product. (-7y - 5)(y + 3) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y 3 -7y -5 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. -7y2 -21y -5y -15 -7y2 + (- 5y -21y ) + (-15) -7y2 -26y - 15

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)

Assignment Multiples of 3 (3,6,9,12,15,….) (rest of paper- extra credit) Page 1 5(2x+6y) Page 2 (x-1)(2x-3) Due Tuesday Quiz Tomorrow Assignment Odd problems Even problems extra credit Due Tuesday

Lesson Quiz 2. (2a3 – a + 3)(a2 + 3a – 5) 1. 5jk(k – 2j) 5jk2 – 10j2k Find each product. 1. 5jk(k – 2j) 5jk2 – 10j2k 2a5 + 6a4 – 11a3 + 14a – 15

x4 –4x3 x2 5x3 5x –2x2 8x –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. x2 –4x 1 x2 5x –2 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 x2 5x3 –20x2 5x –2x2 8x –2 x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2 x4 + x3 – 21x2 + 13x – 2