Chapter 12 Section 4

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

Expand a binomial raised to a power. Objective 1 Expand a binomial raised to a power. Slide 12.4- 3

Expand a binomial raised to a power. Pascal’s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 and so on Slide 12.4- 4

Expand a binomial raised to a power. n factorial (n!) For any positive integer n, By definition, Slide 12.4- 5

Evaluating Factorials CLASSROOM EXAMPLE 1 Evaluating Factorials Evaluate 4!. Solution: Slide 12.4- 6

Evaluating Expressions Involving Factorials CLASSROOM EXAMPLE 2 Evaluating Expressions Involving Factorials Find the value of each expression. Solution: Slide 12.4- 7

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, Slide 12.4- 8

Evaluating Binomial Coefficients CLASSROOM EXAMPLE 3 Evaluating Binomial Coefficients Evaluate Solution: Slide 12.4- 9

Expand a binomial raised to a power. Binomial Theorem For any positive integer n, The binomial theorem can be written in summation notation as Slide 12.4- 10

Using the Binomial Theorem CLASSROOM EXAMPLE 4 Using the Binomial Theorem Expand (x2 + 3)5. Solution: Slide 12.4- 11

Using the Binomial Theorem CLASSROOM EXAMPLE 5 Using the Binomial Theorem Expand Solution: Slide 12.4- 12

Find any specified term of the expansion of a binomial. Objective 2 Find any specified term of the expansion of a binomial. Slide 12.4- 13

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 Slide 12.4- 14

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 Slide 12.4- 15