Holt McDougal Algebra 2

Multiplying Polynomials

Warm UpMultiply.

1. x(x3)

3. 2(5x3)

5. xy(7x2)

6. 3y2(–3y)

7x3y

x4

10x3

–9y3

2. 3x2(x5) 3x7

4. x(6x2) 6x3

Holt McDougal Algebra 2

Multiplying Polynomials

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

Objectives

Holt McDougal Algebra 2

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 2

Multiplying Polynomials

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

Holt McDougal Algebra 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

Multiplying Polynomials

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

Holt McDougal Algebra 2

Multiplying Polynomials

Check It Out! Example 5

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

Expand the expression.

Holt McDougal Algebra 2

Multiplying Polynomials

3. 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

Find each product.

4. Expand the expression. (3a – b)3 27a3 – 27a2b + 9ab2 – b3