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

Holt McDougal Algebra Multiplying Polynomials 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)(a 2 + 4ab + 4b 2 ) Multiply the last two binomial factors. a(a 2 ) + a(4ab) + a(4b 2 ) + 2b(a 2 ) + 2b(4ab) + 2b(4b 2 ) Distribute a and then 2b. a 3 + 4a 2 b + 4ab 2 + 2a 2 b + 8ab 2 + 8b 3 Multiply. a 3 + 6a 2 b + 12ab 2 + 8b 3 Combine like terms.

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 Check It Out! Example 4b Find the product. (2x – 1) 3 (2x – 1)(2x – 1)(2x – 1) Write in expanded form. (2x – 1)(4x 2 – 4x + 1) Multiply the last two binomial factors. 2x(4x 2 ) + 2x(–4x) + 2x(1) – 1(4x 2 ) – 1(–4x) – 1(1) Distribute 2x and then – 1. 8x 3 – 8x 2 + 2x – 4x 2 + 4x – 1 Multiply. 8x 3 – 12x 2 + 6x – 1 Combine like terms.

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