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Copyright © 2014, 2010, 2007 Pearson Education, Inc. Chapter 11 Sequences, Induction, and Probability 11.5 The Binomial Theorem Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1

Objectives: Evaluate a binomial coefficient. Expand a binomial raised to a power.

Definition of a Binomial Coefficient

Example: Evaluating Binomial Coefficients Evaluate each of the following:

Example: Evaluating Binomial Coefficients Evaluate:

A Formula for Expanding Binomials: The Binomial Theorem

Example: Using the Binomial Theorem Expand:

Example: Using the Binomial Theorem (continued) Expand:

Introducing: Pascal’s Triangle Take a moment to copy the first 6 rows. What patterns do you see? Row 5 Row 6

The Binomial Theorem Use Pascal’s Triangle to expand (a + b)5. Use the row that has 5 as its second number. The exponents for a begin with 5 and decrease. 1a5b0 + 5a4b1 + 10a3b2 + 10a2b3 + 5a1b4 + 1a0b5 The exponents for b begin with 0 and increase. In its simplest form, the expansion is a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5. Row 5

The Binomial Theorem Use Pascal’s Triangle to expand (x – 3)4. First write the pattern for raising a binomial to the fourth power. 1 4 6 4 1 Coefficients from Pascal’s Triangle. (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 Since (x – 3)4 = (x + (–3))4, substitute x for a and –3 for b. (x + (–3))4 = x4 + 4x3(–3) + 6x2(–3)2 + 4x(–3)3 + (–3)4 = x4 – 12x3 + 54x2 – 108x + 81 The expansion of (x – 3)4 is x4 – 12x3 + 54x2 – 108x + 81.

The Binomial Theorem For any positive integer, n

Let’s Try Some Expand the following a) (x-y5)3 b) (3x-2y)4

Let’s Try Some Expand the following (x-y5)3

Let’s Try Some Expand the following (3x-2y)4

Let’s Try Some Expand the following (3x-2y)4