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Warm Up Multiply.

Warm Up Multiply. 1. x ( x 3 ). x 4. 2. 3 x 2 ( x 5 ). 3 x 7. 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. 1. x(x3) x4 2. 3x2(x5) 3x7 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. 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.

  6. 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.

  7. 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.

  8. 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

  9. 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)

  10. Check It Out! Example 3 Multiply the two polynomials. 0.02x2+ 0.2x + 3 –0.004x2– 0.1x+ 3 0.06x2 + 0.6x + 9 –0.002x3– 0.02x2 – 0.3x –0.00008x4– 0.0008x3 – 0.012x2 –0.00008x4– 0.0028x3 + 0.028x2+ 0.3x + 9 Mr. Silva’s total manufacturing costs, in thousands of dollars, can be modeled by T(x) = –0.00008x4– 0.0028x3 + 0.028x2+ 0.3x + 9

  11. Example 4: Expanding a Power of a Binomial Find the product. (a + 2b)3 (a + 2b)(a + 2b)(a + 2b) Write in expanded form. (a + 2b)(a2 + 4ab + 4b2) Multiply the last two binomial factors. Distribute a and then 2b. a(a2) + a(4ab)+ a(4b2) + 2b(a2) +2b(4ab) +2b(4b2) a3 + 4a2b + 4ab2 + 2a2b +8ab2 +8b3 Multiply. a3 + 6a2b + 12ab2 + 8b3 Combine like terms.

  12. 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.

  13. This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.

  14. Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k –5)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(k)3(–5)0] + [3(k)2(–5)1] + [3(k)1(–5)2] + [1(k)0(–5)3] k3 – 15k2 + 75k – 125 B. (6m – 8)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(6m)3(–8)0] + [3(6m)2(–8)1] + [3(6m)1(–8)2] + [1(6m)0(–8)3] 216m3 – 864m2 + 1152m – 512

  15. Check It Out! Example 5 Expand each expression. a. (x +2)3 1 3 3 1 Identify the coefficients for n = 3, or row 3. [1(x)3(2)0] + [3(x)2(2)1] + [3(x)1(2)2] + [1(x)0(2)3] x3 + 6x2 + 12x + 8 b. (x –4)5 1 5 10 10 5 1 Identify the coefficients for n = 5, or row 5. [1(x)5(–4)0] + [5(x)4(–4)1] + [10(x)3(–4)2] + [10(x)2(–4)3] + [5(x)1(–4)4] + [1(x)0(–4)5] x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024

  16. Check It Out! Example 5 Expand the expression. c. (3x +1)4 1 4 6 4 1 Identify the coefficients for n = 4, or row 4. [1(3x)4(1)0] + [4(3x)3(1)1] + [6(3x)2(1)2] + [4(3x)1(1)3] + [1(3x)0(1)4] 81x4 + 108x3 + 54x2 + 12x + 1

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