1. Chapter 12
Section 4

2. The Binomial Theorem 12.4 Expand a binomial raised to a power.
Find any specified term of the expansion of a binomial.

3. Expand a binomial raised to a power.
Objective 1
Expand a binomial raised to a power.
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4. Expand a binomial raised to a power. Pascal’s Triangle1
and so on
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5. Expand a binomial raised to a power. n factorial (n!)
For any positive integer n,
By definition,
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6. Evaluating Factorials
CLASSROOM EXAMPLE 1
Evaluating Factorials
Evaluate 4!.
Solution:
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7. Evaluating Expressions Involving Factorials
CLASSROOM EXAMPLE 2
Evaluating Expressions Involving Factorials
Find the value of each expression.
Solution:
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8. Formula for the Binomial Coefficient nCr
Expand a binomial raised to a power.
Formula for the Binomial Coefficient nCr
For nonnegative integers n and r, where r ≤ n,
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9. Evaluating Binomial Coefficients
CLASSROOM EXAMPLE 3
Evaluating Binomial Coefficients
Evaluate
Solution:
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10. Expand a binomial raised to a power. Binomial Theorem
For any positive integer n,
The binomial theorem can be written in summation notation as
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11. Using the Binomial Theorem
CLASSROOM EXAMPLE 4
Using the Binomial Theorem
Expand (x2 + 3)5.
Solution:
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12. Using the Binomial Theorem
CLASSROOM EXAMPLE 5
Using the Binomial Theorem
Expand
Solution:
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13. Find any specified term of the expansion of a binomial.
Objective 2
Find any specified term of the expansion of a binomial.
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14. rth Term of the Binomial Expansion
Find any specified term of the expansion of a binomial.
rth Term of the Binomial Expansion
If n ≥ r – 1, then the rth term of the expansion of (x + y)n is
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15. Finding a Single Term of a Binomial Expansion
CLASSROOM EXAMPLE 6
Finding a Single Term of a Binomial Expansion
Find the fifth term of
Solution:
In the fifth term of the exponent on (–y) is 5 – 1 = 4 and the exponent on is 9 – 4 = 5.
The fifth term is
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