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Transcript
Name
5-7
Class
Date
Reteaching
The Binomial Theorem
You can find the coefficients of a binomial expansion in Pascal’s Triangle.
To create Pascal’s Triangle, start by writing a triangle of 1’s.
This triangle forms the first two rows. Each row has one more
element than the one above it. Each row begins with a 1, and
then each element is the sum of the two closest elements in
the row above. The last element in each row
is a 1.
Pascal’s Triangle
1
2
1
3
6
4
1
1
3
1
1
Problem
1
1
5
1
4
10
1
10
5
1
What is the expansion of (x 1 y)5 ? Use Pascal’s Triangle.
Step 1 The power of the binomial corresponds to the second
number in each row of Pascal’s Triangle. Because the
power of this binomial is 5, use the row of Pascal’s
Triangle with 5 as the second number. The numbers
of this row are the coefficients of the expansion.
Pascal’s Triangle
1
1
1
1
1
1
1
2
3
4
5
1
3
6
10
1
1
4
10
5
1
Step 2 The exponents of the x-terms of the expansion
Count down for x.
begin with the power of the binomial
and decrease to 0. The exponents of
1x 5y 0! 5x 4y1 ! 10x 3y2 ! 10x 2y3 ! 5x 1y 4 ! 1x 0y5
the y-terms of the expansion begin
with 0 and increase to the power of
Count up for y.
the binomial.
Step 3 Simplify all terms to write the expansion in standard form.
(x 1 y)5 5 x5 1 5x4y 1 10x3y2 1 10x2y3 1 5xy4 1 y5
Exercises
Write the expansion of each binomial.
1. (a 1 b)3 a3 1 3a2b 1 3ab2 1 b3
2. (x 2 y)4 x4 2 4x3y 1 6x2y2 2 4xy3 1 y4
3. (r 1 1)5 r5 1 5r4 1 10r3 1 10r2 1 5r 1 1
4. (a 2 b)6 a6 2 6a5b 1 15a4b2 2 20a3b3 1
15a2b4 2 6ab5 1 b6
Prentice Hall Algebra 2 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
69
Name
Class
5-7
Date
Reteaching (continued)
The Binomial Theorem
• The Binomial Theorem states that for any binomial (a 1 b) and any positive integer n,
(a 1 b)n 5 nC0an 1 nC1an21b 1 nC2an22b2 1 . . . 1 nCn21abn21 1 nCnbn.
• The theorem provides an effective method for expanding any power of a binomial.
n!
.
Evaluate the combination nCk as
k!(n 2 k)!
Problem
What is the expansion of (3x 1 2)3 ? Use the Binomial Theorem.
Step 1 Determine a, b, and n.
a 5 3x, b 5 2, n 5 3
Step 2 Use the formula to write the equation.
(3x 1 2)3 5 3C0(3x)3 1 3C1(3x)2(2) 1 3C2(3x)(2)2 1 3C3(2)3
Step 3 Simplify.
5 1(27x3) 1 3(9x2)(2) 1 3(3x)(4) 2 1(8)
5 27x3 1 54x2 1 36x 1 8
Exercises
Fill in the correct coefficients, variables, and exponents for the expanded form
of each binomial.
u 1 ux3y 1 6xuy2 1 uxyu 1 u4 4; 4; 2; 4; 3; y
6. (z 2 y)3 5 zu 2 u z2y 1 u zyu 2 u3 3; 3; 3; 2; y
7. (x 1 z)5 5 xu 1 ux4z 1 10xuz2 1 ux2zu 1 uxz4 1 u5 5; 5; 3; 10; 3; 5; z
5. (x 1 y)4 5 x
Write the expansion of each binomial. Use the Binomial Theorem.
8. (x 1 y)5
9. (x 2 y)5
x5 1 5x4y 1 10x3y2 1 10x2y3 1 5xy4 1 y5
10. (2x 1
y) 3
8x3 1 12x2y 1 6xy2 1 y3
12. (x 2
2y)5
14. (x 2 3y)4
11. (x 1 3y)4
x4 1 12x3y 1 54x2y2 1 108xy3 1 81y4
13. (2x 2 y) 5
32x5 2 80x4y 1 80x3y2 2 40x2y3 1 10xy4 2 y5
15. (4x 2 y)3
x5 2 10x4y 1 40x3y2 2
80x2y3 1 80xy4 2 32y5
x4 2 12x3y 1 54x2y2 2 108xy3 1 81y4
16. (x 2
x5 2 5x4y 1 10x3y2 2 10x2y3 1 5xy4 2 y5
64x3 2 48x2y 1 12xy2 2 y3
1)5
17. (1 2 x)3
x5 2 5x4 1 10x3 2 10x2 1 5x 2 1
18. (x2 1 1) 3 x6 1 3x4 1 3x2 1 1
1 2 3x 1 3x2 2 x3
19. (y2 1 a)4 y8 1 4y6a 1 6y4a2 1 4y2a3 1 a4
Prentice Hall Algebra 2 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
70