Using the Binomial TheoremExpand
= 16 m^4 + 96 m^3 + 216 m^2 + 216 m+ 81 NOW TRY
= 16 m^4 + 418 m^32132 + 614 m^22192 + 412 m 21272 + 81
= 12 m 24 +
4!
1!3!
12 m 23132 +
4!
2!2!
12 m 221322 +
4!
3!1!
12 m 21323 + 34
12 m+ 324
12 m+ 324.
EXAMPLE 4
704 CHAPTER 12 Sequences and Series
Remember:
1 ab 2 m=ambm.Using the Binomial TheoremExpand
=a^5 - NOW TRY
5
2
a^4 b+
5
2
a^3 b^2 -
5
4
a^2 b^3 +
5
16
ab^4 -
1
32
b^5
+ 5 aa
b^4
16
b + a-
b^5
32
b
=a^5 + 5 a^4 a-
b
2
b + 10 a^3 a
b^2
4
b + 10 a^2 a-
b^3
8
b
+
5!
4!1!
aa-
b
2
b
4+ a-
b
2
b
5=a^5 +
5!
1!4!
a^4 a-
b
2
b +
5!
2!3!
a^3 a-
b
2
b
2+
5!
3!2!
a^2 a-
b
2
b
3aa-
b
2
b
5Aa-
b2 B
5.
EXAMPLE 5
Notice that signs
alternate positive
and negative.CAUTION When the binomial is the differenceof two terms, as in Example 5,
the signs of the terms in the expansion will alternate. Those terms with odd exponents
on the second variable expression in Example 5 will be negative, while those
with even exponents on the second variable expression will be positive.
OBJECTIVE 2 Find any specified term of the expansion of a binomial.
Any single term of a binomial expansion can be determined without writing out the
whole expansion. For example, if then the 10th term of has yraised
to the ninth power (since yhas the power of 1 in the second term, the power of 2 in the
third term, and so on). Since the exponents on xand yin any term must have a sum of
n, the exponent on xin the 10th term is The quantities 9 and determine
the factorials in the denominator of the coefficient. Thus, the 10th term of is
n!
9! 1 n- 92!
xn-^9 y^9.
1 x+ y 2 n
n- 9. n- 9
nÚ10, 1 x+y 2 n
A- B
b
2NOW TRY
EXERCISE 4Expand. 1 a+ 3 b 25
NOW TRY ANSWERS
4.
16 y^4x^4
81 -8 x^3 y
27 +8 x^2 y^2
3 -32 xy^3
3 +270 a^2 b^3 + 405 ab^4 + 243 b^5a^5 + 15 a^4 b+ 90 a^3 b^2 +NOW TRY
EXERCISE 5Expand .a
x
3- 2 yb
4rth Term of the Binomial Expansion