Download 10-5 Study Guide and Intervention The Binomial Theorem

NAME
10-5
DATE
PERIOD
Study Guide and Intervention
The Binomial Theorem
Pascal’s Triangle
In Pascal’s triangle, the first and last numbers in
each row is 1 and the number in row 0 is 1. Other numbers are the sum of
the two numbers above them. The first five rows of Pascal’s triangle are
shown below.
Row 0
1
Row 1
1
Row 2
1
Row 3
Row 4
1
1
1
2
3
1
3
4
6
1
4
1
The numbers in Pascal’s triangles are the binomial coefficients when
(a + b)n is expanded. You can use these numbers to expand binomials without
multiplying repeatedly. The first term is an, the last term is bn, and the powers
of a decrease by 1 as the powers of b increase by 1 from left to right.
Example
Use Pascal’s triangle to expand (x + 2y) 5.
First, write the series for (a + b)5 without coefficients. Then replace a with x and b with 2y.
a5b0 + a4b1 + a3b2 + a2b3 + a1b4 + a0b5
Series for (a + b)5
5
0
4
1
3
2
2
3
1
4
0
5
x (2y) + x (2y) + x (2y) + x (2y) + x (2y) + x (2y)
Substitution
5
4
3
2
2
3
4
5
x + x (2y) + x (4y ) + x (8y ) + x (16y ) + 32y
Simplify.
Exercises
Use Pascal’s triangle to expand each binomial.
1. (x + 4)3
2. (3x + y)4
x3 + 12x2 + 48x + 64
3. (7 + g)4
81x4 + 108x3y + 54x2y2 + 12xy3 + y4
2401 + 1372g + 294g2 + 28g3 + g4
4. (m - n)6
m6 - 6m5n + 15m4n2 - 20m3n3 + 15m2n4 - 6mn5 + n6
5. (2a – 2b)5
32a5 - 160a4b + 320a3b2 - 320a2b3 + 160ab4 - 32b5
6. (c + d)7
Chapter 10
c7 + 7c6d + 21c5d2 + 35c4d3 + 35c3d4 + 21c2d5 + 7cd6 + d7
26
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The numbers in the fifth row of Pascal’s triangle are the coefficients.
Following the pattern above, these numbers will be 1, 5, 10, 10, 5, and 1.
1x5 + 5x4 (2y) + 10x3 (4y2) + 10x2 (8y3) + 5x (16y4) + 1 · 32y5
Add coefficients.
5
4
3 2
2 3
4
5
x + 10x y + 40x y + 80x y + 80xy + 32y
Simplify.
NAME
DATE
10-5
Study Guide and Intervention
PERIOD
(continued)
The Binomial Theorem
The Binomial Theorem
The binomial coefficient of the an - r br term in
the expansion of (a + b) is given by nCr. You can find nCr by using a
n
n!
calculator or by finding −
.
(n - r)! r!
The Binomial Theorem states that for any positive integer n, the
expansion of (a + b)n is
C anb0 + nC1 an - 1b1 + nC2 an - 2b2 + … + nCr an - rbr + … + nCn a0bn.
n 0
Example 1
Find the coefficient of the fourth term in the
expansion of (5a + 2b)6.
For (5a + 2b)6 to have the form (a + b)n, let a = 5a and b = 2b. Since r
increases from 0 to n, r is one less than the number of the term. Evaluate 6C3.
6!
6!
· 5 · 4 · 3!
C3 = −
=−
= 6−
or 20
6
(6 - 3)!3!
3!3!
3!3!
Example 2
Use the Binomial Theorem to expand (3x + 7)4.
Let a = 3x and b = 7.
(3x + 7)4 = 4C0(3x)4(7)0 + 4C1(3x)3(7)1 + 4C2(3x)2(7)2 + 4C3(3x)1(7)3 + 4C4(3x)0(7)4
= 1 · 81x4 · 1 + 4 · 27x3 · 7 + 6 · 9x2 · 49 + 4 · 3x · 343 + 1 · 1 · 2401
= 81x4 + 756x3 + 2646x2 + 4116x + 2401
Exercises
Find the coefficient of the indicated term in each expansion.
1. (x + 5)6, fourth term
2. (3a + 4b)8, a3b5 term
2500
1,548,288
Use the Binomial Theorem to expand each binomial.
3. (x + 3)5
4. (4x + 2y)3
x5 + 15x4 + 90x3 + 270x2
+ 405x + 243
5. (x - 2y)4
64x3 + 96x2y + 48xy2 + 8y3
6. (2x - 3y)4
x4 - 8x3y + 24x2y2 - 32xy3 + 16y4
Chapter 10
16x4 - 96x3y + 216x2y2 - 216xy3 +
81y4
27
Glencoe Precalculus
Lesson 10-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
The binomial coefficient of the fourth term in (a + b)6 is 20. Substitute for a
and b in an - rbr.
20(5a)6 – 3(2b)3 = 20(5a)3(2b)3
= 20(125a3)(8b)
= 20,000a3b
The coefficient is 20,000.