The Binomial Theorem Use Pascal’s Triangle to expand the following. 1.
(x + 2y)5 = x 5 + 10x 4 y + 40x 3 y 2 + 80x 2 y 3 + 80xy 4 + 32y 5
2.
Algebra II Section 6.8 (3x − 4y)7 = 2187x 7 − 20412x 6 y + 81648x 5 y 2 − 181440x 4 y 3 + 241920x 3 y 4 − 193536x 2 y 5
+86016xy 6 − 16384y 7
Use the Binomial Theorem to expand each binomial. 3.
(5x − 7y)6 = 15625x 6 − 131250x 5 y + 459375x 4 y 2 − 857500x 3 y 3 + 900375x 2 y 4 − 504210xy 5 + 117649y 6
4.
(x + 4y) = x + 20x y + 160x y + 640x y + 1280x y + 1024y
Answer the following. 5. A family has five children. Assume that the probability of having a boy is 0.5. Write the term in the expansion of (b + g)5 for each outcome described. Then evaluate each probability. a.) exactly 3 boys b.) exactly 4 boys c.) exactly 4 girls 3
2
4
1
1
4
5 C2 (0.5) (0.5) ≈ 0.3125
5 C1 (0.5) (0.5) ≈ 0.15625
5 C 4 (0.5) (0.5) ≈ 0.15625
2
5
10
8
or binompdf (5, 0.5, 3)
6. 7. 8. 9. 6 2
4
3
2
4
5
or binompdf (5, 0.5, 4)
or binompdf (5, 0.5, 4)
In the expansion of (m + n)9 , one of the terms contains m 3 . a.) b.) What is the exponent of n in this term? n = 6 What is the coefficient of this term? 84 Suppose 8 C3 x 5 y 3 is a term of a binomial expansion. Write the next term. 70x 4 y 4 The term 126c 4 d 5 appears in the expansion of (c + d)n . Find n. n=9 The coefficient of the second term in the expansion of (r + s)n is 7. Find the value of n, and write the complete term. n = 7; Term: 7 C1 (r)6 (s)1 = 7r 6 s Find the specified term of each binomial expansion. 10. Third term of (x + 3)12 594x10 11. Fourth term of (x + 2)5 80x 2 12. Twelfth term of (2 + x)11 x11 13. Seventh term of (x − 2y)6 64y 6 14. Seventh term of (x 2 − 2y)11 29568x10 y 6 15. Eighth term of (x 2 + y 2 )13 1716x12 y14