* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Section 9.5 The Binomial Theorem
Survey
Document related concepts
Wiles's proof of Fermat's Last Theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Vincent's theorem wikipedia , lookup
System of polynomial equations wikipedia , lookup
Elementary mathematics wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Lattice model (finance) wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Transcript
• Section 9.5 The Binomial Theorem 163 Name______________________________________________ Section 9.5 The Binomial Theorem Objective: In this lesson you learned how to use the Binomial Theorem and Pascal’s Triangle to calculate binomial coefficients and binomial expansions. Important Vocabulary Define each term or concept. Binomial coefficients The coefficients of a binomial expansion. I. Binomial Coefficients (Pages 683−684) What you should learn How to use the Binomial Theorem to calculate binomial coefficients List four general observations about the expansion of ( x + y ) n for various values of n. 1) In each expansion, there are n + 1 terms. 2) In each expansion, x and y have symmetrical roles. The powers of x decrease by 1 in successive terms, whereas the powers of y increase by 1. 3) The sum of the powers in each term is n. 4) The coefficients increase and then decrease in a symmetric pattern. The Binomial Theorem states that in the expansion of (x + y)n = xn + nxn – 1y + . . . + nCrxn – ryr + . . . + nxyn – 1 + yn, the coefficient of xn – ryr is nCr = n!/((n − r)!r!) Example 1: Find the binomial coefficient 792 . 12 C 5 . II. Pascal’s Triangle (Page 685) Pascal’s Triangle is a triangular pattern, named for French mathematician Blaise Pascal, in which the first and last numbers in each row are 1 and every other number in each row is formed by . . . adding the two numbers immediately above the number. The numbers in Pascal’s triangle are precisely the same numbers that are the coefficients of binomial expansions . Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved. What you should learn How to use Pascal’s Triangle to calculate binomial coefficients 164 Chapter 9 • Sequences, Series, and Probability Construct rows 0 through 6 of Pascal’s Triangle. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 III. Binomial Expansions (Pages 686−687) Writing out the coefficients for a binomial that is raised to a power is called expanding a binomial . What you should learn How to use binomial coefficients to write binomial expansions Example 2: Use the binomial coefficients from the appropriate row of Pascal’s Triangle to expand ( x + 2) 5 x5 + 10x4 + 40x3 + 80x2 + 80x + 32 Additional notes Homework Assignment Page(s) Exercises Larson/Hostetler Precalculus/Precalculus with Limits Notetaking Guide IAE Copyright © Houghton Mifflin Company. All rights reserved.