Download 1. Expand (a b)n Using Pascal`s Triangle Section 15.4 The Binomial

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Chapter 15 Sequences and Series
Section 15.4 The Binomial Theorem
Objectives
1.
2.
3.
4.
5.
Expand (a ⴙ b)n
Using Pascal’s
Triangle
Evaluate Factorials
n
Evaluate a b
r
Expand (a ⴙ b)n
Using the Binomial
Theorem
Find a Specified Term
in the Expansion of
(a ⴙ b)n
In this section, we will learn how to expand a binomial, (a b) n, where n is a nonnegative
integer. We first encountered expansion of binomials in Chapter 6 when we learned to
expand binomials such as (a b) 2 and (a b) 3. To expand (a b) 2 means to find the
product (a b)(a b). Multiplying the binomials using FOIL gives us
(a b) 2 (a b)(a b)
a2 ab ab b2
a2 2ab b2
Multiply using FOIL.
To expand (a b) 3 means to find the product (a b)(a b)(a b) or
(a b) (a b) 2. Expanding (a b) 3 gives us
(a b) 3 (a b)(a b)(a b)
(a b)(a b) 2
(a b)(a2 2ab b2 )
a3 2a2b ab2 a2b 2ab2 b3
a3 3a2b 3ab2 b3
Distribute.
Combine like terms.
Expanding, for example, (a b) 5 in this way would be an extremely long process. There
are other ways to expand binomials, and the first one we will discuss is Pascal’s triangle.
1. Expand (a ⴙ b)n Using Pascal’s Triangle
Here are the expansions of (a b) n for several values of n:
(a b) 0 1
(a b) 1 a b
(a b) 2 a2 2ab b2
(a b) 3 a3 3a2b 3ab2 b3
(a b) 4 a4 4a3b 6a2b2 4ab3 b4
(a b) 5 a5 5a4b 10a3b2 10a2b3 5ab4 b5
Notice the following patterns in the expansion of (a b) n:
1) There are n 1 terms in the expansion of (a b) n. For example, in the expansion of
(a b) 4, n 4 and the expansion contains 4 1 5 terms.
2) The first term is an and the last term is bn.
3) Reading the expansion from left to right, the exponents on a decrease by 1 from one
term to the next, while the exponents on b increase by 1 from one term to the next.
4) In each term in the expansion, the sum of the exponents of the variables is n.
The coefficients of the terms in the expansion follow a pattern too. If we write the coefficients in triangular form, we obtain Pascal’s triangle, named after seventeenth-century
French mathematician Blaise Pascal. The numbers in the nth row of the triangle tell us the
coefficients of the terms in the expansion of (a b) n.
Coefficients of the Terms
in the Expansion of:
(a b) :
(a b) 1:
(a b) 2:
(a b) 3:
(a b) 4:
(a b) 5:
Pascal’s Triangle
0
1
1
1
1
1
2
3
1
3
1
1 4 6 4 1
1 5 10 10 5 1
etc.
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Notice that the first and last numbers of each row in the triangle are 1. The other numbers
in the triangle are obtained by adding the two numbers above it. For example, here is how
to obtain the sixth row from the fifth:
Fifth row (n 4):
1 4 6 4 1
Sixth row (n 5): 1 5 10 10 5 1
Example 1
Expand (a b) 6.
Solution
The coefficients of the terms in (a b) 5 are given by the last row of the triangle above.
We must find the next row of the triangle to find the coefficients of the terms in the
expansion of (a b) 6.
(a b) 5:
1
5 10 10 5 1
(a b) 6: 1 6 15 20 15 6 1
Recall that the first term is an and the last term is bn. Since n 6, the first term will be
a6, and the exponent of a will decrease by 1 for each term. The variable b will appear in
the second term and increase by 1 for each term until the last term, b6.
(a b) 6 a6 6a5b 15a4b2 20a3b3 15a2b4 6ab5 b6
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You Try 1
Expand (a b) 7.
Although Pascal’s triangle is a better way to expand (a b) n than doing repeated polynomial multiplication, it can be tedious for large values of n. A more practical way to expand
a binomial is by using the binomial theorem. Before learning this method, we need to learn
about factorials and binomial coefficients.
2. Evaluate Factorials
The notation n! is read as “n factorial.”
Definition
n! n(n 1)(n 2)(n 3) . . . (1) , where n is a positive integer.
Note
By definition, 0! 1.
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Example 2
Evaluate.
a)
4!
b)
7!
Solution
a) 4! 4 ⴢ 3 ⴢ 2 ⴢ 1 24
b) 7! 7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1 5040
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You Try 2
Evaluate.
a)
3!
b)
6!
n
3. Evaluate a b
r
Factorials are used to evaluate binomial coefficients. A binomial coefficient has the form
n
a b, read as “the number of combinations of n items taken r at a time” or as “n choose r.”
r
n
a b is used extensively in many areas of mathematics including probability. Another
r
n
notation for a b is nCr.
r
Definition
Binomial Coefficient
n
n!
a b
r
r !(n r)!
where n and r are positive integers and r n.
Example 3
Evaluate.
5
a) a b
3
9
b) a b
2
3
c) a b
3
4
d) a b
0
Solution
5
a) To evaluate a b, substitute 5 for n and 3 for r.
3
n
n!
a b
r
r!(n r)!
5
5!
a b
3
3!(5 3)!
5!
3!2!
5ⴢ4ⴢ3ⴢ2ⴢ1
(3 ⴢ 2 ⴢ 1)(2 ⴢ 1)
Let n 5 and r 3.
Subtract.
Rewrite each factorial as a product.
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At this point, do not find the products in the numerator and denominator. Instead,
divide out common factors in the numerator and denominator.
5ⴢ4ⴢ3ⴢ2ⴢ1
(3 ⴢ 2 ⴢ 1)(2 ⴢ 1)
20
2
10
Divide out common factors.
Multiply.
Simplify.
5
a b 10
3
9
b) To evaluate a b, substitute 9 for n and 2 for r.
2
9
9!
a b
Let n 9 and r 2.
2
2!(9 2)!
9!
Subtract.
2! 7!
9ⴢ8ⴢ7ⴢ6ⴢ5ⴢ4ⴢ3ⴢ2ⴢ1
Rewrite each factorial as a product.
(2 ⴢ 1)(7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1)
9ⴢ8ⴢ7ⴢ6ⴢ5ⴢ4ⴢ3ⴢ2ⴢ1
Divide out common factors.
(2 ⴢ 1)(7 ⴢ 6 ⴢ 5 ⴢ 4 ⴢ 3 ⴢ 2 ⴢ 1)
72
2
36
9
a b 36
2
3
c) To evaluate a b, substitute 3 for n and for r.
3
n
n!
a b
r!(n r)!
r
3
3!
a b
Let n 3 and r 3.
3
3!(3 3)!
3!
Subtract.
3! 0!
3!
Divide out common factors; 0! 1.
3!(1)
1
Simplify.
1
1
3
a b1
3
4
d) To evaluate a b, substitute 4 for n and 0 for r.
0
4
4!
Let n 4 and r 0.
a b
0
0!(4 0)!
4!
Subtract.
0! 4!
4!
Divide out common factors; 0! 1.
(1)(4!)
1
1
Simplify.
1
4
a b1
0
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Note
We can extend the results of c) and d) and say that for any natural number n,
n
n
a b 1 and a b 1
n
0
You Try 3
Evaluate.
a)
4
a b
1
b)
8
a b
5
c)
3
a b
0
6
d) a b
6
4. Expand (a ⴙ b)n Using the Binomial Theorem
Now that we can evaluate a binomial coefficient, we state the binomial theorem for expanding (a b) n.
Definition
Binomial Theorem: For any positive integer n,
n
n
n
n
b abn1 bn
(a b) n an a b an1b a b an2b2 a b an3b3 . . . a
1
2
3
n1
The same patterns that emerged in the expansion of (a b) n using Pascal’s triangle appear
when using the binomial theorem. Keep in mind that
1) there are n 1 terms in the expansion.
2) the first term in the expansion is an and the last term is bn.
3) after an, the exponents on a decrease by 1 from one term to the next, while b is introduced in the second term and then the exponents on b increase by 1 from one term to
the next.
4) in each term in the expansion, the sum of the exponents of the variables is n.
Example 4
Use the binomial theorem to expand (a b) 4.
Solution
Let n 4 in the binomial theorem.
4
4
4
(a b) 4 a4 a b a41b a b a42b2 a b a43b3 b4
1
2
3
4
4
4
a4 a b a3b a b a2b2 a b ab3 b4
1
2
3
Notice that the exponents of a decrease by 1 while the exponents of b increase by 1.
4! 3
4! 2 2
4!
ab
ab ab3 b4
1! 3!
2! 2!
3! 1!
a4 4a3b 6a2b2 4ab3 b4
a4 This is the same result as the expansion on p. 918.
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You Try 4
Use the binomial theorem to expand (a b) 3.
Example 5
Use the binomial theorem to expand (x 6) 3.
Solution
Substitute x for a, 6 for b, and 3 for n in the binomial theorem to expand (x 6) 3.
3
3
(x 6) 3 (x) 3 a b(x) 31 (6) a b(x) 32 (6) 2 (6) 3
1
2
x3 3(x2 )(6) (3)x(36) 216
x3 18x2 108x 216
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You Try 5
Use the binomial theorem to expand (y 5) 4.
When expanding a binomial containing the difference of two terms, rewrite the expression in terms
of addition.
Example 6
Use the binomial theorem to expand (2x 3y) 5.
Solution
Since the binomial theorem applies to the expansion of (a b) n, rewrite (2x 3y) 5 as
[2x (3y)] 5.
Substitute 2x for a, 3y for b, and 5 for n in the binomial theorem. Be sure to put 2x and
3y in parentheses to find the expansion correctly.
5
5
[2x (3y)] 5 (2x) 5 a b(2x) 51 (3y) a b12x) 52 (3y) 2
1
2
5
5
53
3
a b(2x)
(3y) a b(2x) 54 (3y) 4 (3y) 5
3
4
5
4
32x (5)(2x) (3y) (10)(2x) 3 (9y2 ) (10)(2x) 2 (27y3 )
(5)(2x) 1 (81y4 ) (243y5 )
32x5 (5)(16x4 )(3y) (10)(8x3 )(9y2 )
(10)(4x2 )(27y3 ) (5)(2x)(81y4 ) (243y5 )
5
32x 240x4y 720x3y2 1080x2y3 810xy4 243y5
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You Try 6
Use the binomial theorem to expand (3x 4y) 4.
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5. Find a Specified Term in the Expansion of (a ⴙ b)n
If we want to find a specific term of a binomial expansion without writing out the entire
expansion, we can use the following formula.
Definition
The kth Term of a Binomial Expansion: The kth term of the expansion of (a b) n is given by
n!
ank1bk1
(n k 1)!(k 1)!
where k n 1.
Example 7
Find the fifth term in the expansion of (c2 2d) 8.
Solution
Since we want to find the fifth term, k 5, use the formula above with a c2,
b 2d, n 8, and k 5. The fifth term is
8!
8!
(c2 ) 851 (2d) 51 (c2 ) 4 (2d) 4
(8 5 1)!(5 1)!
4! 4!
70c8 (16d 4 )
1120c8d 4
You Try 7
Find the sixth term in the expansion of (2m n2 ) 9.
Using Technology
We will discuss how to compute factorials and the binomial coefficient on a graphing calculator.
Sometimes it is quicker to calculate them by hand, and sometimes a calculator will make our work
easier.
Evaluating 3! can be done very easily by multiplying: 3! 3 2 1 6. To find 10! by hand we
would multiply: 10! 10 9 8 7 6 5 4 3 2 1 3,628,800. On a graphing
calculator, we could find 10! either by performing this multiplication or we can use a special function.
Graphing calculators have a factorial key built in. It is found using the
MATH key. When you press MATH , move the arrow over to the
PRB column, and you will see this menu:
Notice that choice 4 is the factorial symbol.
To compute 10!, enter 10 and then press MATH . Highlight PRB so
that you see the screen at above right. Choose 4: ! and press ENTER .The
screen displays 10!. Press ENTER to see that 10! 3,628,800.
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n
Because the binomial coefficient, a b, is used so often in mathematical applications, most graphing
r
calculators have a built-in key that performs the calculations for you. It is located on the same menu
as the factorial. Refer to the first calculator screen to see that nCr is choice number 3.
9
To find the value of a b, press 9 MATH , and then highlight PRB.
4
Choose 3: nCr and press ENTER . Now enter 4 and press
ENTER . The screen will look like the next screen here.The value
9
of a b is 126.
4
Although the calculator has functions to evaluate factorials and binomial coefficients, sometimes it is
actually quicker to evaluate them by hand. Think about this as you evaluate the following problems.
Evaluate each of the following using the methods discussed in this section. Verify the result using a
graphing calculator. Think about which method you prefer for each problem.
1)
4!
2)
6!
5)
7
a b
4
6) a
3)
15
b
8
5
4) a b
2
9!
7) a
18
b
14
8) a
25
b
24
Answers to You Try Exercises
1)
a7 7a6b 21a5b2 35a4b3 35a3b4 21a2b5 7ab6 b7
2)
a) 6
5)
y4 20y3 150y2 500y 625
7)
b) 720
3)
a) 4
b) 56
c) 1
d) 1
4) a3 3a2b 3ab2 b3
81x4 432x3y 864x2y2 768xy3 256y4
6)
4 10
2016m n
Answers to Technology Exercises
1)
24
2)
720
3)
362,880
4)
10
5)
35
6)
6435
7)
3060
8)
25
15.4 Exercises
Objective 1: Expand (a ⴙ b)n Using Pascal’s Triangle
Objective 2: Evaluate Factorials
9) In your own words, explain how to evaluate n! for any
positive integer.
1) In your own words, explain how to construct Pascal’s
triangle.
10) Evaluate 0!.
2) What are the first and last terms in the expansion of
(a b) n?
Evaluate.
Use Pascal’s triangle to expand each binomial.
3) (r s)3
4) (m n)4
5) (y z)5
6) (c d)6
7) (x 5)4
8) (k 2)5
VIDEO
11) 2!
12) 3!
13) 5!
14) 6!
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Objective 3: Evaluate a b
n
r
Evaluate each binomial coefficient.
5
15) a b
2
4
16) a b
2
7
VIDEO 17) a b
3
8
18) a b
5
19) a
9
20) a b
3
10
b
4
44) (4c 3d) 4
45) (x2 1) 3
46) (w3 2) 3
4
1
47) a m 3nb
2
5
1
48) a a 2bb
3
3
1
49) a y 2z2 b
3
50) at 2 Find the indicated term of each binomial expansion.
9
21) a b
7
22) a
4
23) a b
4
5
24) a b
5
52) ( y 4) 7; fifth term
6
25) a b
1
3
26) a b
1
54) (z 3) 9; seventh term
5
27) a b
0
7
28) a b
0
11
b
8
51) (k 5) 8; third term
53) (w 1) 15; tenth term
VIDEO
55) (q 3) 9; second term
56) (u 2) 7; fourth term
57) (3x 2) 6; fifth term
58) (2w 1) 9; seventh term
29) How many terms are in the expansion of (a b) ?
9
30) Before expanding (t 4) using the binomial theorem,
how should the binomial be rewritten?
6
59) (2y2 z) 10; eighth term
60) (p 3q2 ) 8; fifth term
61) (c3 3d2 ) 7; third term
62) (2r3 s4 ) 6; sixth term
Use the binomial theorem to expand each expression.
63) (5u v3 ) 11; last term
31) ( f g)
64) (4h k4 ) 12; last term
32) (c d)
3
33) (w 2)
5
34) (h 4)
4
4
35) (b 3) 5
36) (t 9) 3
37) (a 3) 4
38) ( p 2) 3
39) (u v)
40) ( p q)
3
41) (3m 2)
4
1 4
ub
2
Objective 5: Find a Specified Term in the Expansion
of (a ⴙ b) n
Objective 4: Expand (a ⴙ b) n Using the
Binomial Theorem
VIDEO
43) (3a 2b) 5
5
42) (2k 1) 4
n
65) Show that a b 1 for any positive integer n.
n
n
66) Show that a b n for any positive integer n.
1