10-6 The Binomial Theorem
Expand each binomial.
1. (c + d)
5
SOLUTION: Replace n with 5 in the Binomial Theorem.
6
3. (x – 4)
SOLUTION: Replace n with 6 in the Binomial Theorem. 5
5. (x + 3)
SOLUTION: Replace n with 5 in the Binomial Theorem.
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10-6 The Binomial Theorem
5
5. (x + 3)
SOLUTION: Replace n with 5 in the Binomial Theorem.
7. GENETICS If a woman is equally as likely to have a baby boy or a baby girl, use binomial expansion to determine
the probability that 5 of her 6 children are girls. Do not consider identical twins.
SOLUTION: 6
Find the probability by expanding (g + b) .
By adding the coefficients of the polynomial, we determine that there are 64 combinations of girls and boys.
5
6g b represents the number of combinations with 5 girls and 1 boy.
Therefore, the probability that 5 of her 6 children are girls is 0.09375.
Find the indicated term of each expression.
9. fifth term of (x + 3y)
8
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
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For the fifth term k = 4.
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the probability
10-6Therefore,
The Binomial
Theoremthat 5 of her 6 children are girls is 0.09375.
Find the indicated term of each expression.
9. fifth term of (x + 3y)
8
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the fifth term k = 4.
11. sixth term of (2c – 3d)
8
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the sixth term k = 5.
13. first term of (3a + 8b)
5
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the first term k = 0.
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10-6 The Binomial Theorem
13. first term of (3a + 8b)
5
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the first term k = 0.
Expand each binomial.
15. (a – b)
6
SOLUTION: Replace n with 6 in the Binomial Theorem.
6
17. (x + 6)
SOLUTION: Replace n with 6 in the Binomial Theorem.
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10-6 The Binomial Theorem
6
17. (x + 6)
SOLUTION: Replace n with 6 in the Binomial Theorem.
19. (2a + 4b)
4
SOLUTION: Replace n with 4 in the Binomial Theorem.
21. COMMITTEES If an equal number of men and women applied to be on a community planning committee and the
committee needs a total of 10 people, find the probability that 7 of the members will be women. Assume that
committee members will be chosen randomly.
SOLUTION: 10
Find the probability by expanding (m + w) .
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By adding the coefficients of the polynomial, we determine that there are 1024 combinations of men and women.
3 7
120m w represents the number of combinations with 7 women and 3 men.
10-6 The Binomial Theorem
21. COMMITTEES If an equal number of men and women applied to be on a community planning committee and the
committee needs a total of 10 people, find the probability that 7 of the members will be women. Assume that
committee members will be chosen randomly.
SOLUTION: 10
Find the probability by expanding (m + w) .
By adding the coefficients of the polynomial, we determine that there are 1024 combinations of men and women.
3 7
120m w represents the number of combinations with 7 women and 3 men.
Therefore, the probability that 7 of the members are women is
.
Find the indicated term of each expression.
23. third term of (x + 2z)
7
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the third term k = 2.
25. seventh term of (2a – 2b)
8
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the seventh term k = 6.
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10-6 The Binomial Theorem
25. seventh term of (2a – 2b)
8
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the seventh term k = 6.
9
27. fifth term of (x – 4)
SOLUTION: Use the Binomial Theorem to write the expansion in sigma notation.
For the fifth term k = 4.
Expand each binomial.
29. SOLUTION: Replace n with 5 in the Binomial Theorem.
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10-6 The Binomial Theorem
Expand each binomial.
29. SOLUTION: Replace n with 5 in the Binomial Theorem.
31. SOLUTION: Replace n with 5 in the Binomial Theorem.
33. CCSS SENSE-MAKING In
, let p represent the likelihood of a success and q represent the
likelihood of a failure.
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a. If a place-kicker makes 70% of his kicks within 40 yards, find the likelihood that he makes 9 of his next 10
attempts from within 40 yards.
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10-6 The Binomial Theorem
33. CCSS SENSE-MAKING In
, let p represent the likelihood of a success and q represent the
likelihood of a failure.
a. If a place-kicker makes 70% of his kicks within 40 yards, find the likelihood that he makes 9 of his next 10
attempts from within 40 yards.
b. If a quarterback completes 60% of his passes, find the likelihood that he completes 8 of his next 10 attempts.
c. If a team converts 30% of their two-point conversions, find the likelihood that they convert 2 of their next 5
conversions.
SOLUTION: a. Substitute n = 10, k = 9, p = 0.7 and q = 0.3 in
.
b. Substitute n = 10, k = 8, p = 0.6 and q = 0.4 in
.
c. Substitute n = 5, k = 2, p = 0.3 and q = 0.7 in
.
n n
35. REASONING Explain how the terms of (x + y) and (x – y) are the same and how they are different.
SOLUTION: n
Sample answer: While they have the same terms, the signs for (x + y) will all be positive, while the signs for (x –Page
y) 9
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n
will alternate.
10-6 The Binomial Theorem
n n
35. REASONING Explain how the terms of (x + y) and (x – y) are the same and how they are different.
SOLUTION: n
Sample answer: While they have the same terms, the signs for (x + y) will all be positive, while the signs for (x – y)
n
will alternate.
4
37. OPEN ENDED Write a power of a binomial for which the second term of the expansion is 6x y.
SOLUTION: Sample answer:
39. PROBABILITY A desk drawer contains 7 sharpened red pencils, 5 sharpened yellow pencils, 3 unsharpened red
pencils, and 5 unsharpened yellow pencils. If a pencil is taken from the drawer at random, what is the probability that
it is yellow, given that it is one of the sharpened pencils?
A
B
C
D
SOLUTION: Favorable outcomes: {5 sharpened yellow pencils}
Possible outcomes: {7 sharpened red pencils, 5 sharpened yellow pencils}
Probability A is the correct option.
41. GEOMETRY Christie has a cylindrical block that she needs to paint for an art project.
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Probability is the
correct option.
10-6AThe
Binomial
Theorem
41. GEOMETRY Christie has a cylindrical block that she needs to paint for an art project.
What is the surface area of the cylinder in square inches rounded to the nearest square inch? F 1960
G 2413
H 5127
J 6634
SOLUTION: The radius of the cylinder is 12 in.
Substitute 20 for h and 12 for r in the formula to find the volume V of the cylinder.
G is the correct option.
Find the first five terms of each sequence.
43. a 1 = –2, a n + 1 = a n + 5
SOLUTION: The Manual
first five
termsbyofCognero
the sequence
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are –2, 3, 8, 13 and 18.
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is the
correct option.
10-6GThe
Binomial
Theorem
Find the first five terms of each sequence.
43. a 1 = –2, a n + 1 = a n + 5
SOLUTION: The first five terms of the sequence are –2, 3, 8, 13 and 18.
45. a 1 = 4, a n + 1 = 3a n – 6
SOLUTION: The first five terms of the sequence are 4, 6, 12, 30 and 84.
Find the sum of each infinite geometric series, if it exists.
47. SOLUTION: Find the value of r to determine if the sum exists.
eSolutions
by exists.
Cognero
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the sum
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five terms
of the sequence are 4, 6, 12, 30 and 84.
10-6The
Thefirst
Binomial
Theorem
Find the sum of each infinite geometric series, if it exists.
47. SOLUTION: Find the value of r to determine if the sum exists.
Since
, the sum exists.
Use the formula to find the sum.
49. TRAVEL A trip between two towns takes 4 hours under ideal conditions. The first 150 miles of the trip is on an
interstate, and the last 130 miles is on a highway with a speed limit that is 10 miles per hour less than on the
interstate.
a. If x represents the speed limit on the interstate, write expressions for the time spent at that speed and for the time
spent on the other highway.
b. Write and solve an equation to find the speed limits on the two highways.
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SOLUTION: a.
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10-6 The Binomial Theorem
49. TRAVEL A trip between two towns takes 4 hours under ideal conditions. The first 150 miles of the trip is on an
interstate, and the last 130 miles is on a highway with a speed limit that is 10 miles per hour less than on the
interstate.
a. If x represents the speed limit on the interstate, write expressions for the time spent at that speed and for the time
spent on the other highway.
b. Write and solve an equation to find the speed limits on the two highways.
SOLUTION: a.
Distance traveled on the interstate is 150 miles.
So, the time spent on the interstate is
.
Distance traveled on highways is 130 miles.
Speed on highways is x – 10.
So, the time spent on the highways is
.
b.
Total time taken for the trip is 4 hrs.
.
Solve for x.
So, the speed on the interstate is 75 mph.
Thus, the speed on the highway is 75 – 10 = 65 mph.
State whether each statement is true or false when n = 1. Explain.
51. 3n + 5 is even.
SOLUTION: Substitute n = 1 in the expression and simplify.
Since 8 is an even number, given statement is true for n = 1.
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So, the speed on the interstate is 75 mph.
the speed on
the highway is 75 – 10 = 65 mph.
10-6Thus,
The Binomial
Theorem
State whether each statement is true or false when n = 1. Explain.
51. 3n + 5 is even.
SOLUTION: Substitute n = 1 in the expression and simplify.
Since 8 is an even number, given statement is true for n = 1.
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