14.3%the%binomial%theorem%% -...

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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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Page 1: 14.3%The%Binomial%Theorem%% - Pre-Calculusprecalculus.flippedmath.com/uploads/1/1/3/0/11305589/pc... · 2020-03-10 · 5) 4th term in expansion of (2x - 1)3 6)1st term in expansion

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!

!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%!+ ! !!!!=%

Page 2: 14.3%The%Binomial%Theorem%% - Pre-Calculusprecalculus.flippedmath.com/uploads/1/1/3/0/11305589/pc... · 2020-03-10 · 5) 4th term in expansion of (2x - 1)3 6)1st term in expansion

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%

⋮%

Page 3: 14.3%The%Binomial%Theorem%% - Pre-Calculusprecalculus.flippedmath.com/uploads/1/1/3/0/11305589/pc... · 2020-03-10 · 5) 4th term in expansion of (2x - 1)3 6)1st term in expansion

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

!! !!!!!!

!

!!!%%%% (! + !) ! =

Page 4: 14.3%The%Binomial%Theorem%% - Pre-Calculusprecalculus.flippedmath.com/uploads/1/1/3/0/11305589/pc... · 2020-03-10 · 5) 4th term in expansion of (2x - 1)3 6)1st term in expansion

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)  

Page 5: 14.3%The%Binomial%Theorem%% - Pre-Calculusprecalculus.flippedmath.com/uploads/1/1/3/0/11305589/pc... · 2020-03-10 · 5) 4th term in expansion of (2x - 1)3 6)1st term in expansion

14.3%The%Binomial%Theorem%% 6

Write your questions anthoughts here!