14.3%the%binomial%theorem%% -...
TRANSCRIPT
14.3%The%Binomial%Theorem%% 1Write 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! !! =!nCk!= ! !!!! !!! ! 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!
!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%
14.3%The%Binomial%Theorem%% 2Write your questions and thoughts here!
EXAMPLES:%
Find%each%term%described.%
1. The!third!term!in!the!expansion!(y!+!5)4 2. The!third!term!in!the!expansion!(2x!K!3)6
Expand%completely.%
3. (2!+!3y)4
Pascal’s%Triangle%
!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%
⋮%
!%!!+ !!%
!!! + !!"+ !!!%!!! + !!!!+ !!!! + !!!%
!!! + !!!!+ !!!!! + !!"! + !!!%!!! + !!!!+ !"!!!! + !"!!!! + !!"! + !!!%
⋮%
Looking%at%
just%the%
coefficients:%
!%!!!!!!!!%
!!!!!!!!!!!!!%!!!!!!!!!!!!!!!!!!!!!%
!!!!!!!!!!!!!!!!!!!!!!!!!!!%!!!!!!!!!!!!!"!!!!!"!!!!!!!!!!!%
⋮%
Which%
happens%to%
be%the%same%
as:%
0C0
1C0%%%%1C1
2C0%%%%%%2C1%%%%%2C2%
3C0%%%%%%3C1%%%%%3C2%%%%%%%%3C3%
4C0%%%%%%4C1%%%%%4C2%%%%%%%%4C3%%%%%%%4C4%
5C0%%%%%%5C1%%%%%5C2%%%%%%%%5C3%%%%%%%5C4%%%%%%5C5%
⋮%
14.3%The%Binomial%Theorem%% 3Write your questions anthoughts here!
Practice%14.3%
Evaluate!each!combination.!Use!the!formula!and!then!check!your!answers!with!your!calculator.!
1. 11C5 2. 13C8 3. 10C6 4. 10C4
Find each term described.
5) 4th term in expansion of (2x - 1)3 6) 1st term in expansion of (4x - 2y)4 7) 3rd term in expansion of (y + 3)4
8) 2nd term in expansion of (2u + v)4 9) 5th term in expansion of (x + 2y)4 10) 3rd term in expansion of (y - 5x)3
Expand completely.
11) (2n - 1)3 12) (1 + 3y)3 13) (x - 4y)4
14) (x + 4)415) (y - 3x)5
!! !!!!!!
!
!!!%%%% (! + !) ! =
Application 14.3%The%Binomial%Theorem%% 4Write your questions an
thoughts here!
1. Expand!completely:! !(7x!–!3y)5 2. Evaluate:!!!13C0
3.! a.! Complete!Pascal’s!Triangle!
b. Add!across!the!rows.!What!pattern!do!you!find?
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)
14.3%The%Binomial%Theorem%% 6
Write your questions anthoughts here!