Multiplying Polynomials Warm Up Multiply 1 xx 3

Multiplying Polynomials 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 Holt Mc. Dougal Algebra 2

Multiplying Polynomials Objectives Multiply polynomials. Use binomial expansion to expand binomial expressions that are raised to positive integer powers. Holt Mc. Dougal Algebra 2

Multiplying Polynomials To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents. Holt Mc. Dougal Algebra 2

Multiplying Polynomials Example 1: Multiplying a Monomial and a Polynomial Find each product. A. 4 y 2(y 2 + 3) 4 y 2 + 4 y 2 3 4 y 4 + 12 y 2 Distribute. Multiply. B. fg(f 4 + 2 f 3 g – 3 f 2 g 2 + fg 3) fg f 4 + fg 2 f 3 g – fg 3 f 2 g 2 + fg 3 Distribute. f 5 g + 2 f 4 g 2 – 3 f 3 g 3 + f 2 g 4 Multiply. Holt Mc. Dougal Algebra 2

Multiplying Polynomials Check It Out! Example 1 Find each product. a. 3 cd 2(4 c 2 d – 6 cd + 14 cd 2) 3 cd 2 4 c 2 d – 3 cd 2 6 cd + 3 cd 2 14 cd 2 12 c 3 d 3 – 18 c 2 d 3 + 42 c 2 d 4 Distribute. Multiply. b. x 2 y(6 y 3 + y 2 – 28 y + 30) x 2 y 6 y 3 + x 2 y y 2 – x 2 y 28 y + x 2 y 30 6 x 2 y 4 + x 2 y 3 – 28 x 2 y 2 + 30 x 2 y Holt Mc. Dougal Algebra 2 Distribute. Multiply.

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 Mc. Dougal Algebra 2

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

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

Multiplying Polynomials Example 2 B: Multiplying Polynomials Find the product. (y 2 – 7 y + 5)(y 2 – y – 3) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. y 2 –y – 3 The top left corner is the first 4 3 2 y –y – 3 y term in the product. Combine terms along diagonals to get – 7 y 3 7 y 2 21 y the middle terms. The bottom right corner is the last term in 5 2 5 y – 15 the product. y 4 + (– 7 y 3 – y 3 ) + (5 y 2 + 7 y 2 – 3 y 2) + (– 5 y + 21 y) – 15 y 4 – 8 y 3 + 9 y 2 + 16 y – 15 Holt Mc. Dougal Algebra 2

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

Multiplying Polynomials Check It Out! Example 2 b Find the product. (x 2 – 4 x + 1)(x 2 + 5 x – 2) Multiply each term of one polynomial by each term of the other. Use a table to organize the products. x 2 – 4 x 1 The top left corner is the first 4 3 2 x – 4 x x term in the product. Combine terms along diagonals to get 2 5 x 5 x 3 – 20 x 5 x the middle terms. The bottom right corner is the last term in – 2 x 2 8 x – 2 the product. x 4 + (– 4 x 3 + 5 x 3) + (– 2 x 2 – 20 x 2 + x 2) + (8 x + 5 x) – 2 x 4 + x 3 – 21 x 2 + 13 x – 2 Holt Mc. Dougal Algebra 2

Multiplying Polynomials Example 4: Expanding a Power of a Binomial Find the product. (a + 2 b)3 (a + 2 b)(a + 2 b) Write in expanded form. (a + 2 b)(a 2 + 4 ab + 4 b 2) Multiply the last two binomial factors. Distribute a and then 2 b. a(a 2) + a(4 ab) + a(4 b 2) + 2 b(a 2) + 2 b(4 ab) + 2 b(4 b 2) a 3 + 4 a 2 b + 4 ab 2 + 2 a 2 b + 8 ab 2 + 8 b 3 a 3 + 6 a 2 b + 12 ab 2 + 8 b 3 Holt Mc. Dougal Algebra 2 Multiply. Combine like terms.

Multiplying Polynomials Check It Out! Example 4 a Find the product. (x + 4)4 (x + 4)(x + 4) Write in expanded form. (x + 4)(x 2 + 8 x + 16) Multiply the last two binomial factors. (x 2 + 8 x + 16) Multiply the first two binomial factors. Distribute x 2 and then 8 x and then 16. x 2(x 2) + x 2(8 x) + x 2(16) + 8 x(x 2) + 8 x(8 x) + 8 x(16) + 16(x 2) + 16(8 x) + 16(16) Multiply. x 4 + 8 x 3 + 16 x 2 + 8 x 3 + 64 x 2 + 128 x + 16 x 2 + 128 x + 256 x 4 + 16 x 3 + 96 x 2 + 256 x + 256 Holt Mc. Dougal Algebra 2 Combine like terms.

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

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 Mc. Dougal Algebra 2

Multiplying Polynomials This information is formalized by the Binomial Theorem, which you will study further in Chapter 11. Holt Mc. Dougal Algebra 2

Multiplying Polynomials Example 5: Using Pascal’s Triangle to Expand Binomial Expressions Expand each expression. A. (k – 5)3 1331 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 – 15 k 2 + 75 k – 125 B. (6 m – 8)3 1331 Identify the coefficients for n = 3, or row 3. [1(6 m)3(– 8)0] + [3(6 m)2(– 8)1] + [3(6 m)1(– 8)2] + [1(6 m)0(– 8)3] 216 m 3 – 864 m 2 + 1152 m – 512 Holt Mc. Dougal Algebra 2

Multiplying Polynomials Check It Out! Example 5 Expand each expression. a. (x + 2)3 Identify the coefficients for n = 3, or row 3. 1331 [1(x)3(2)0] + [3(x)2(2)1] + [3(x)1(2)2] + [1(x)0(2)3] x 3 + 6 x 2 + 12 x + 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] x 5 – 20 x 4 + 160 x 3 – 640 x 2 + 1280 x – 1024 Holt Mc. Dougal Algebra 2

Multiplying Polynomials Check It Out! Example 5 Expand the expression. c. (3 x + 1)4 14641 Identify the coefficients for n = 4, or row 4. [1(3 x)4(1)0] + [4(3 x)3(1)1] + [6(3 x)2(1)2] + [4(3 x)1(1)3] + [1(3 x)0(1)4] 81 x 4 + 108 x 3 + 54 x 2 + 12 x + 1 Holt Mc. Dougal Algebra 2

Multiplying Polynomials Lesson Quiz Find each product. 1. 5 jk(k – 2 j) 5 jk 2 – 10 j 2 k 2. (2 a 3 – a + 3)(a 2 + 3 a – 5) 2 a 5 + 6 a 4 – 11 a 3 + 14 a – 15 3. The number of items is modeled by 0. 3 x 2 + 0. 1 x + 2, and the cost per item is modeled by g(x) = – 0. 1 x 2 – 0. 3 x + 5. Write a polynomial c(x) that can be used to model the total cost. – 0. 03 x 4 – 0. 1 x 3 + 1. 27 x 2 – 0. 1 x + 10 4. Find the product. (y – 5)4 y 4 – 20 y 3 + 150 y 2 – 500 y + 625 5. Expand the expression. (3 a – b)3 27 a 3 – 27 a 2 b + 9 ab 2 – b 3 Holt Mc. Dougal Algebra 2