Binomial Expansion Pascal Triangle U2

Hon Alg 2: Unit 2 Powers of Binomials: Binomial Expansion Examples: Use FOIL as needed (x + y)0 = __________________________ (x + y)3 = _____________________________________ (x + y)1 = __________________________ (x + y)2 = __________________________ (n + 4)2 = __________________________ (n + 1)3 = _____________________________________ (c – 3)2 = __________________________ 2 (2a + 3b) = __________________________ (c – 2)3 = _____________________________________ Based on the power of the binomial, do you notice any pattern or relationship between the terms of your polynomial solution and the original terms of the binomial? Common multiples for each term depending on the power PASCAL’S TRIANGLE ● What patterns do you notice? Add the two previous diagonal terms from the row above ● Use your pattern to find the next two rows. ● What do you notice any relationship between the power answers and Pascal’s Triangle? Each row represents the coefficients for the overall power EACH Diagonal/ Term of Row = specific power of first term Use Pascal’s Triangle to expand each of the following: Write powers of first term counting down from n and write powers of second term count up from 0 1. (x + y)4 = 1x4 + 4x3y + 6x2y2 + 4x1y3 + 1y4 2. (m + n)5 m5 + 5m4n + 10m3n2 + 10 m2n2 + 5mn4 + 1n5 3. (x – 2)4 4. (2x – 3)6 1x4 + 4x3(-2) + 6x2(-2)2 + 4x1(-2)3 + 1(-2)4 x4 – 8x3 + 24x2 – 32x + 16 Combination Statement: or alternative notation Total number of ways to select an unordered group of k items from a total of n items ● FACTORIAL (N!): The product of consecutive integers from N to 1. 1) 3! = 3*2*1 = 6 2) 5! = 5*4*3*2*1 = 120 3) 6! = 6*5*4*3*2*1 = 720 4) 5) 7) 8) 6) 9) 10) Based on #7 – 10: Combination = values of Pascal’s Triangle: n= row and k = diagonal or term in row Binomial Theorem: The binomial expansion is based on the summation of combination statements and varying powers of your binomial terms. (be careful with negative signs) Hint #1: Powers of each summation term will add to equal power of binomial expression (n) Hint #2: Combinations will always be paired with the power of the second term from the binomial (b) Hint #3: Summation from 0 to n means there will be one more term than the power of the binomial o (… + … )1 = 2 terms o (… + … )2 = 3 terms o (… + … )3 = 4 terms Examples: Expand using the binomial theorem. 1. (x + y)3 x3 + 3x2y + 3xy2 + y3 2. (3x + y)4 81x4 + (4)27x3y + (6)9x2y2 + (4)3xy3 + y4 = 81x4 + 108x3y + 54x2y2 + 12xy3 + y4 3. (2x + y)5 (2x)5 + 5(2x)4y+ 10(2x) 3y2 + 10(2x) 2y3 + 5(2x)y4 + y5 = 32x5 + 90x4y + 80x3y2 + 40x2y3 + 10xy4 + y5