14.3 The Binomial Theorem 1 Write your questions and thoughts here! The Binomial Theorem Expansion, the boring and tedious way: The AMAZING Binomial Theorem: ! + ! ! = ! ! ! !!! ! ! ! !!! where ! ! is said “n choose k” and represents the number of ways to select k things from n. where ! ! = n C k = !! !! !!! ! and n!= n(n – 1)(n – 2)…(2)1 EXAMPLES: Evaluate: ! ! ! ! ! ! Binomial Expansion, using the amazing Binomial Theorem: ! + ! ! = ! ! ! !!! ! ! ! !!! (a + b) 3 (2x – 3y) 3 You try! Expand completely. (5x + 3y) 4 ! + ! ! = ! + ! ! = ! + ! ! = ! + ! ! = ! + ! ! = ! + ! ! =
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14.3%The%Binomial%Theorem%% 1Write your questions and thoughts here!
c. Write!the!pattern!as!a!geometric!sequence!forthe!sum!of!the!nth!row.!!For!simplicity,assume!the!top!row!is!row!0,!or!n!=!0.
Use!an!=!____________
d. Find!the!sum!of!the!20th!row!ofPascals!triangle.
e. That was too easy!!!Find!the!sum!of!the!first!100!rows!of!Pascal’s!triangle!
f. Find!the!Fibonacci!sequence!in!Pascal’s!triangle.
g. Sully!loves!to!color.!!On!the!next!page,!color!over!each!number!that!is!even.!What!pattern!doyou!see?!!!(Boo-yah! You’ve just been Sierpinskied!)
!! = !! !!!!!
!!! !!
Skillz Review! Write the equation of a line with the given slope that passes through the given point. In slope-‐intercept form: y = mx + b In point-‐slope form: y -‐ y1 = m(x – x1) 1. slope = 𝟑𝟓; through (-‐15,-‐3)
2. slope = 0; through (7,6) 3. slope = 2; through (4,-‐1) 4. slope = − 𝟏𝟖; through ( ,π)