College Algebra 9th txtbk

SECTION 8–6 The Binomial Formula 561
Technology Connections
The table feature on a graphing calculator provides an efficient
alternative to calculating the coefficients of Example 2
one by one (Fig. 3).
Z Figure 3 y 1 C 4,x 3 4x (2) x .
MATCHED PROBLEM 2 Use the binomial formula to expand (2m 5n) 3 .
ZZZ EXPLORE-DISCUSS 3
(A) Compute each term and also the sum of the alternating series
a 6 0 b a6 1 b a6 2 b . . . a 6 6 b
(B) What result about an alternating series can be deduced by letting a 1 and
b 1 in the binomial formula?
EXAMPLE 3 Using the Binomial Formula
Find the term containing x 9 in the expansion of (x 3) 14 .
SOLUTION
In the expansion
(x 3) 14 a
14
k 0
a 14
k b x14k 3 k
the exponent of x is 9 when k 5. So the term containing x 9 is
a 14 5 b x9 3 5 (2,002)(243)x 9 486,486x 9
MATCHED PROBLEM 3 Find the term containing y 8 in the expansion of (2 y) 14 .
EXAMPLE 4 Using the Binomial Formula
If the terms in the expansion of (x 2) 20 are arranged in decreasing powers of x, find the
fourth term and the sixteenth term.
SOLUTION
In the expansion of (a b) n , the exponent of b in the rth term is r 1 and the exponent
of a is n (r 1). Therefore
Fourth term:
a 20 3 b x17 (2) 3
Sixteenth term:
a 20
15 b x5 (2) 15
20 19 18
20 19 18 17 16
x 17 (8) x 5 (32,768)
3 2 1
5 4 3 2 1
9,120x 17 508,035,072 x 5