Holt McDougal Algebra Multiplying Polynomials Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers. Objectives

Holt McDougal Algebra Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

Holt McDougal Algebra Multiplying Polynomials Find each product. Example 1: Multiplying a Monomial and a Polynomial A. 4y 2 (y 2 + 3) Distribute. B. fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) 4y 2  y 2 + 4y 2  3 4y 2 (y 2 + 3) Multiply. 4y y 2 fg(f 4 + 2f 3 g – 3f 2 g 2 + fg 3 ) Distribute. Multiply. fg  f 4 + fg  2f 3 g – fg  3f 2 g 2 + fg  fg 3 f 5 g + 2f 4 g 2 – 3f 3 g 3 + f 2 g 4

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

Holt McDougal Algebra Multiplying Polynomials (y 2 – 7y + 5)(y 2 – y – 3) Find the product. Example 2B: Multiplying Polynomials Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y4y4 –y3–y3 –3y 2 –7y 3 7y27y2 21y 5y25y2 –5y–15 y 2 –y –3 y2y2 –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. y 4 + (–7y 3 – y 3 ) + (5y 2 + 7y 2 – 3y 2 ) + (–5y + 21y) – 15 y 4 – 8y 3 + 9y y – 15

Holt McDougal Algebra Multiplying Polynomials (x 2 – 4x + 1)(x 2 + 5x – 2) Find the product. Check It Out! Example 2b Multiply each term of one polynomial by each term of the other. Use a table to organize the products. x4x4 –4x 3 x2x2 5x35x3 –20x 2 5x5x –2x 2 8x8x–2 x 2 –4x 1 x2x2 5x5x –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. x 4 + (–4x 3 + 5x 3 ) + ( –2 x 2 – 20x 2 + x 2 ) + (8x + 5x) – 2 x 4 + x 3 – 21x x – 2

Holt McDougal Algebra Multiplying Polynomials Check It Out! Example 4a Find the product. (x + 4) 4 (x + 4)(x + 4)(x + 4)(x + 4) Write in expanded form. Multiply the last two binomial factors. (x + 4)(x + 4)(x 2 + 8x + 16) Multiply the first two binomial factors. (x 2 + 8x + 16)(x 2 + 8x + 16) x 2 (x 2 ) + x 2 (8x) + x 2 (16) + 8x(x 2 ) + 8x(8x) + 8x(16) + 16(x 2 ) + 16(8x) + 16(16) Distribute x 2 and then 8x and then 16. Multiply. x 4 + 8x x 2 + 8x x x + 16x x Combine like terms. x x x x + 256

Holt McDougal Algebra 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 Multiplying Polynomials This information is formalized by the Binomial Theorem, which you will study further in Chapter 11.

Holt McDougal Algebra Multiplying Polynomials Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5) 3 B. (6m – 8) 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 ] k 3 – 15k k – 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 ] 216m 3 – 864m m – 512

Holt McDougal Algebra Multiplying Polynomials Check It Out! Example 5 Expand each expression. a. (x + 2) 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 ] x 3 + 6x x + 8 b. (x – 4) 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 ] x 5 – 20x x 3 – 640x x – 1024

Holt McDougal Algebra Multiplying Polynomials 4. Find the product. (y – 5) 4 Lesson Quiz 2. (2a 3 – a + 3)(a 2 + 3a – 5) 5jk 2 – 10j 2 k 1. 5jk(k – 2j) 2a 5 + 6a 4 – 11a a – 15 y 4 – 20y y 2 – 500y –0.03x 4 – 0.1x x 2 – 0.1x The number of items is modeled by 0.3x x + 2, and the cost per item is modeled by g(x) = –0.1x 2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost. Find each product. 5. Expand the expression. (3a – b) 3 27a 3 – 27a 2 b + 9ab 2 – b 3