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4.2 Pascal’s Triangle and the Binomial Theorem

4.2 Pascal’s Triangle and the Binomial Theorem. Consider the binomial expansions again…. That is, there are ways to get that term. specifically…. Consider the x 2 a term. There are 3 ways to get that term. Pascal’s Triangle using. Value of n 0 1 2 3 4 5. r = 0. r = 1.

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4.2 Pascal’s Triangle and the Binomial Theorem

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  1. 4.2 Pascal’s Triangle and the Binomial Theorem

  2. Consider the binomial expansions again…

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

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

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

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

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

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

  9. Pascal’s Identity

  10. General Term of Binomial Expansion The general in the expansion of (a + b)nis

  11. Example 3 Use Pascal’s Identity to write an expression for n = 47 r = 14

  12. 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!

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