The Binomial Theorem Honors PreCalc 10 5 Pascals

The Binomial Theorem Honors Pre-Calc 10. 5

Pascal’s Triangle • An arrangement/pattern of numbers adding numbers in row above it. 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1

Pascal’s Triangle to help expand binomials in form (a +b)n (a+b)0 (a+b)1 (a+b)2 (a+b)3 (a+b)4 (a+b)5 1 a 0 b 0 1 a 1 b 0 1 a 0 b 1 1 a 2 b 0 2 a 1 b 1 1 a 0 b 2 1 a 3 b 0 3 a 2 b 1 3 a 1 b 2 1 a 0 b 3 1 a 4 b 0 4 a 3 b 1 6 a 2 b 2 4 a 1 b 3 1 a 0 b 4 1 a 5 b 0 5 a 4 b 1 10 a 3 b 2 10 a 2 b 3 5 a 1 b 4 1 a 0 b 5

Example 1: Use Pascal’s Triangle to expand (2 x + 3 y)5 (2 x – 7)3

The Binomial Theorem • While Pascal’s triangle would work to expand any (a+b)n expression, when n gets too large that process would take a long time. The Binomial Theorem and additional formulas with it , can help shorten the process.

Formula for the Binomial Coefficients of (a +b)n • The binomial coefficient of any an-rbr term in the expansion of (a+b)n is n. Cr • (a+b)3 = 3 C 0 a 3 b 0 + 3 C 1 a 2 b 1 + 3 C 2 a 1 b 2 + 3 C 3 a 0 b 3

Example 2: Find Binomial Coefficients • Find the coefficient of the indicated term in each expression (a+b)7; 5 th term (a – b)13; 3 rd term

Example 3: Binomials w/ Coefficients other than 1 • Find the coefficient of the x 7 y 2 term in the expansion of (4 x – 3 y)9 • Find the coefficient of the x 3 y 5 term in the expansion of (2 x – 3 y)8

The Binomial Theorem • For any positive integer n, the expansion of (a + b)n is given by (a+b)n = n. C 0 anb 0 + n. C 1 an-1 b 1 + n. C 2 an-2 b 2 + … + n. Cran-rbr + … + n. Cna 0 bn

Example 4: Expand the binomial using the Binomial Theorem (3 x – y)4 (5 m+4)3