In this general expression, remember to start with the exponent on y, which is 1 less
than the term number r. Then subtract that exponent from nto get the exponent on x:
The two exponents are then used as the factorials in the denominator of
the coefficient.
Finding a Single Term of a Binomial ExpansionFind the fourth term of the expansion of
In the fourth term, 2bhas an exponent of and ahas an exponent of
The fourth term is determined as follows.
Let.Simplify the factorials.= 960 a^7 b^3 Multiply. NOW TRY
= 120 a^718 b^32
= n=10, x=a, y= 2 b, r= 4
10 # 9 # 8
3 # 2 # 11 a^7218 b^32
10!
3!7!
1 a^7212 b 23
10 - 3 =7.
4 - 1 = 3
1 a+ 2 b 210.
EXAMPLE 6
n- 1 r- 12.
SECTION 12.4 The Binomial Theorem 705
Parentheses MUST
be used for 2b.NOW TRY
EXERCISE 6
Find the sixth term of the ex-
pansion of 12 m-n^228.Complete solution available
on the Video Resources on DVD
12.4 EXERCISES
Evaluate each expression. See Examples 1–3.
1.6! 2.4! 3.8! 4.9! 5.Use the binomial theorem to expand each expression. See Examples 4 and 5.Write the first four terms of each binomial expansion. See Examples 4 and 5.Find the indicated term of each binomial expansion. See Example 6.- ; fourth term 34. ; fifth term
- ; seventh term 36. ; eighth term
- ; third term 38. ; fourth term
39.The middle term of 40.The middle term of41.The term with x^9 y^4 in 13 x^3 - 4 y^225 42.The term with x^8 y^2 in 12 x^2 + 3 y 261 x^2 + 2 y 26 1 m^3 + 2 y 281 k- 129 1 r- 4211aa+b
3b15
ax+y
2b812 m+n 210 1 a+ 3 b 21212 p- 3 q 211 1 t^2 +u^22101 x^2 +y^22151 r+ 2 s 212 1 m+ 3 n 220 13 x-y 2141 x^2 + 124 1 y^3 + 224 13 x^2 - y^22312 p^2 - q^223ax
3- 2 yb
5
ax
2- yb
4
12 x+ 323 14 x+ 2231 m+n 24 1 x+r 25 1 a-b 25 1 p-q 246 C 2 7 C 4 13 C 11 13 C 2
4!# 5 6!# 7
5!
5!0!
4!
0!4!
7!
3!4!
6!
4!2!
NOW TRY ANSWER
6.- 448 m^3 n^10