4.2 Pascal’s Triangle and the Binomial Theorem

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Presentation transcript:

4.2 Pascal’s Triangle and the Binomial Theorem

Consider the binomial expansions again…

specifically… Consider the x2a term There are 3 ways to get that term. That is, there are ways to get that term.

Pascal’s Triangle using Value of n 1 2 3 4 5 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5

Binomial Theorem The coefficients of the form are called binomial coefficients.

Expand and simplify using the binomial theorem (x + y)6 (2x – 1)4

Expand and simplify using the binomial theorem (3x – 2y)5

Example 2 Using the binomial theorem, rewrite 1 + 10x2 + 40x4 + 80x6 + 80x8 + 32x10 in the form (a + b)n. n = 5 (6 terms)

Pascal’s Identity

General Term of Binomial Expansion The general in the expansion of (a + b)n is

Example 3 Use Pascal’s Identity to write an expression for n = 47

Example 4 Consider the expansion of What is the constant term? or We want an-rbr = x0 8 – 3r = 0 r must be a whole number, so there is no constant term!