13.5 The Binomial Theorem

There are several theorems and strategies that allow us to expand binomials raised to powers such as (x + y)4 or (2x – 5y)7.

One of these is Pascal’s Triangle

1 1 11 2 1

1 3 3 11 4 6 4 1

1 5 10 10 5 11 6 15 20 15 6 1

• each row starts & ends with a 1• to get each term, you add the two numbers diagonally above that spot

*we can continue like this forever!

your turn:again:

Another is the idea of factorials5! 5 4 3 2 1 ! ( 1) ( 2) ... (2) (1)

! !

( )! ! ( )! !n r

n n n n

nn nC

rn r r n r r

*remember it can also be written as

Ex 1) Find the lie and fix it!

A) B) C)10 1610!

362,880 210 524,1606 1110

No 4368

There is a connection between the numbers in Pascal’s Triangle andTake for instance Row 4 1 4 6 4 1notice that

This helps us both expand binomials as well as find a particular term of an expansion without expanding the whole thing.

4 4 4 4 4

0 1 2 3 4

n

r

How to expand a binomial: (a + b)n

(1) Coefficients: use Pascal’s Triangle or

(2) Powers of each variable: the powers on the first term descend from n …. 0

the powers on the second term ascend from 0 …. n

Ex 2) Using Pascal’s Triangle, expand (x + y)4

1 x4 y0 + 4 x3 y1 + 6 x2 y2 + 4 x1 y3 + 1 x0 y4

x4 + 4x3y + 6x2y2 + 4xy3 + y4

n

r

Ex 3) Expand

52

0 1 2 35 4 3 22 2 2 2

4 51 02 2

5 4 2 3 4 2 6 8 10

2

1 2 5 2 10 2 10 2

5 2 1 2

32 80 80 40 10

x y

x y x y x y x y

x y x y

x x y x y x y xy y

Ex 4) Find the coefficient of the indicated term & identify the missingexponent.

x?y9 ; (x + y)11

x2y9

*each set of powers adds to the total of 11 ? + 9 = 11

? = 2this is simply

2 9

total!

exponent!exponent!

11!55 55

2!9!x y

Ex 5) Find the term involving the specified variable.b6 in (a – b)14

8 6 8 614!3003

8!6!a b a b

Ex 6) Find the indicated term of the expansion.a) the third term of (a + 5b)4

b) the fourth term of (2a – 6b)11

*the third term would have a2 (count down)

2 2 2 2 2 24!5 6 25 150

2!2!a b a b a b

*fourth term a11 a10 a9 a8

8 3 8 3 8 311!2 6 165 256 216 9123840

8!3!a b a b a b

Homework

#1305 Pg 712 #1–9 odd, 13, 17, 19, 21, 23–27, 32, 33