Holt McDougal Algebra 2

6-2 Multiplying Polynomials

Multiply polynomials.Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

Objectives

Holt McDougal Algebra 2

6-2 Multiplying Polynomials

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

Holt McDougal Algebra 2

6-2 Multiplying Polynomials

Find each product.

Example 1: Multiplying a Monomial and a Polynomial

A. 4y2(y2 + 3)

Distribute.

B. fg(f4 + 2f3g – 3f2g2 + fg3)

4y2 y2 + 4y2 3

4y2(y2 + 3)

Multiply.4y4 + 12y2

fg(f4 + 2f3g – 3f2g2 + fg3)Distribute.

Multiply.

fg f4 + fg 2f3g – fg 3f2g2 + fg fg3

f5g + 2f4g2 – 3f3g3 + f2g4

Holt McDougal Algebra 2

6-2 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 2

6-2 Multiplying Polynomials

Find the product.

Example 2A: Multiplying Polynomials

(a – 3)(2 – 5a + a2)

a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)

Method 1 Multiply horizontally.

a3 – 5a2 + 2a – 3a2 + 15a – 6 a3 – 8a2 + 17a – 6

Write polynomials in standard form.Distribute a and then –3.

Multiply. Add exponents.

Combine like terms.

(a – 3)(a2 – 5a + 2)

Holt McDougal Algebra 2

6-2 Multiplying Polynomials

(y2 – 7y + 5)(y2 – 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.

y4 –y3 –3y2

–7y3 7y2 21y

5y2 –5y –15

y2 –y –3y2

–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.

y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15 y4 – 8y3 + 9y2 + 16y – 15

Holt McDougal Algebra 2

6-2 Multiplying Polynomials

(x2 – 4x + 1)(x2 + 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.

x4 –4x3 x2

5x3 –20x2 5x

–2x2 8x –2

x2 –4x 1x2

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

Holt McDougal Algebra 2

6-2 Multiplying PolynomialsCheck 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)(x2 + 8x + 16)

Multiply the first two binomial factors.

(x2 + 8x + 16)(x2 + 8x + 16)

x2(x2) + x2(8x) + x2(16) + 8x(x2) + 8x(8x) + 8x(16) + 16(x2) + 16(8x) + 16(16)

Distribute x2 and then 8x and then 16.

Multiply.x4 + 8x3 + 16x2 + 8x3 + 64x2 + 128x + 16x2 + 128x + 256

Combine like terms.x4 + 16x3 + 96x2 + 256x + 256

Holt McDougal Algebra 2

6-2 Multiplying PolynomialsNotice 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 2

6-2 Multiplying Polynomials

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

Holt McDougal Algebra 2

6-2 Multiplying PolynomialsExample 5: Using Pascal’s Triangle to Expand Binomial

ExpressionsExpand each expression. A. (k – 5)3

B. (6m – 8)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

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

Holt McDougal Algebra 2

6-2 Multiplying PolynomialsCheck 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

Holt McDougal Algebra 2

6-2 Multiplying Polynomials

4. Find the product. (y – 5)4

Lesson Quiz

2. (2a3 – a + 3)(a2 + 3a – 5) 5jk2 – 10j2k1. 5jk(k – 2j)

2a5 + 6a4 – 11a3 + 14a – 15

y4 – 20y3 + 150y2 – 500y + 625

–0.03x4 – 0.1x3 + 1.27x2 – 0.1x + 10

3. The number of items is modeled by 0.3x2 + 0.1x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 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 27a3 – 27a2b + 9ab2 – b3