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Multiplying Polynomials
Warm Up
Multiply.
1. x(x3) x4
2. 3x2(x5) 3x7
3. 2(5x3) 10x3
4. x(6x2) 6x3
5. xy(7x2)
7x3y
6. 3y2(–3y)
–9y3
Holt McDougal Algebra 2
Multiplying Polynomials
Objectives
Use binomial expansion to expand
binomial expressions that are raised to
positive integer powers.
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 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]
k3 – 15k2 + 75k – 125
B. (6m – 8)3
1331
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
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]
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
Expand the expression.
c. (3x + 1)4
14641
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
Holt McDougal Algebra 2
Multiplying Polynomials
Lesson Quiz
Find each product.
1. 5jk(k – 2j) 5jk2 – 10j2k
2. (2a3 – a + 3)(a2 + 3a – 5)
2a5 + 6a4 – 11a3 + 14a – 15
3. Find the product.
(y – 5)4 y4 – 20y3 + 150y2 – 500y + 625
4. Expand the expression.
(3a – b)3 27a3 – 27a2b + 9ab2 – b3
Holt McDougal Algebra 2