College Algebra 9th txtbk

SECTION 8–6 The Binomial Formula 563
Because the right side of the last equation is the right side of P k1 , we have shown that
P k 1 follows from P k .
CONCLUSION
P n is true. That is, the binomial formula holds for all positive integers n.
ANSWERS TO MATCHED PROBLEMS
1. x 5 5x 4 10x 3 10x 2 5x 1 2. 8m 3 60m 2 n 150mn 2 125n 3
3. 192,192y 8 4. 3,060u 14 ; 31,824u 7
8-6 Exercises
1. What is a binomial?
2. What is a binomial coefficient?
3. Explain how the entries in Pascal’s triangle are generated.
4. How can Pascal’s triangle be used to expand (a b) 5 ?
In Problems 5–12, use Pascal’s triangle to evaluate each
expression.
5. a8 6. 7. 8. a 9 3 b
a8 4 b
a9 6 b
7 b
9. C 7,5 10. C 7,3 11. C 9,0 12. C 10 , 10
In Problems 13–20, evaluate each expression.
a
(2x 11 ; x 6
13. 3 b 14. 9 b 15. 4 b 16.
12
11 b
33. (x 1) 7 ; x 4 34. (x 1) 8 ; x 5 35. 1)
17. C 52,3 18. C 52,4 19. C 12,6 20. C 12,11
Expand Problems 21–32 using the binomial formula.
21. (m n) 3 22. (x 2) 3 23. (2x 3y) 3
24. (3u 2v) 3 25. (x 2) 4 26. (x y) 4
27. (m 3n) 4 28. (3p q) 4 29. (2x y) 5
30. (2x 1) 5 31. (m 2n) 6 32. (2x y) 6
In Problems 33–42, find the term of the binomial expansion
containing the given power of x.
36. (3x 1) 12 ; x 7 37. (2x 3) 18 ; x 14 38. (3x 2) 17 ; x 5
39. (x 2 1) 6 ; x 8 40. (x 2 1) 9 ; x 7 41. (x 2 1) 9 ; x 11
42. (x 2 1) 10 ; x 14
In Problems 43–50, find the indicated term in each expansion if
the terms of the expansion are arranged in decreasing powers of
the first term in the binomial.
43. (u v) 15 ; seventh term 44. (a b) 12 ; fifth term
45. (2m n) 12 ; eleventh term 46. (x 2y) 20 ; third term
47. [(w2) 2] 12 ; seventh term 48. (x 3) 10 ; fourth term
49. (3x 2y) 8 ; sixth term 50. (2p 3q) 7 ; fourth term
In Problems 51–54, use the binomial formula to expand and
simplify the difference quotient
f (x h) f (x)
h
for the indicated function f. Discuss the behavior of the simplified
form as h approaches 0.
51. f(x) x 3 52. f(x) x 4
53. f(x) x 5 54. f(x) x 6
In Problems 55–58, use a graphing calculator to graph each
sequence and to display it in table form.
55. Find the number of terms of the sequence
a 20 0 b, a20 1 b, a20 b, . . . , a20
2 20 b
that are greater than one-half of the largest term.
56. Find the number of terms of the sequence
a 40 0 b, a40 1 b, a40 2
b, . . . , a40
40 b
that are greater than one-half of the largest term.
57. (A) Find the largest term of the sequence a 0 , a 1 , a 2 , . . . , a 10 to
three decimal places, where
a k a 10 k b (0.6)10k (0.4) k
(B) According to the binomial formula, what is the sum of the
series a 0 a 1 a 2 . . . a 10 ?
58. (A) Find the largest term of the sequence a 0 , a 1 , a 2 , . . . , a 10 to
three decimal places, where
a k a 10 k b (0.3)10k (0.7) k