Ch. 8 – Sequences, Series, and Probability 8.5 – The Binomial Theorem

Expand the following expressions Expand the following expressions. Leave your answer in decreasing powers of x. (x + y)0 = (x + y)1 = (x + y)2 = (x + y)3 = (x + y)4 =

Let’s examine the coefficients of each power of expansion of (x + y)n : Find (x + y)5. Find (x + y)6. This is Pascal’s Triangle!

Rules of Pascal’s Triangle: First and last # in each row is 1 An entry is the sum of the two entries above it Its entries are the coefficients of binomial expansions Each row is symmetric Pascal’s Triangle may be useful to find expansion coefficients for single digit powers

Ex: Expand (x + 2)3 . Ex: Expand (2x + 3)4. Use Pascal’s Triangle, but note that final coefficients are not the same as before… Ex: Expand (2x + 3)4.

Find the 3rd term of the expansion of (x + y)7.

Find the 2nd term of the expansion of (x – 4)3.

Find the 4th term of the expansion of (x2 + 2)6.

Mathematically, the entries in Pascal’s Triangle are called binomial coefficients and can be found as follows: For the expansion of (x + y)n, the coefficient of the xn-ryr term is You will also see this written as , pronounced “n choose r” Ex: Find 8C2 by hand. This means that if you expanded (x + y)8, the coefficient of the x6y2 or x2y6 term would be 28

Ex: Find and . Fact:

Evaluate by hand. 1 2 15 10 20

Evaluate by hand. 66 300 330 165 792