11.8 The General Binomial Expansion

Recall

New notation:

!

( )! !n r

nC

n r r

n

r

0

0

Use Calc:1

1

0

1

1

1 1

2

0

2

1

1 22

2

1

3

0

3

1

1 33

2

33

3

1

4

0

4

1

1 44

2

64

3

44

4

1

Pas

cal’s

3 0 2 1 1 2 0 33! 3! 3! 3!

3!0! 2!1! 1!2! 0!3!a b a b a b a b

Ex 1) Expand (a + b)3

3 3 0 2 1 1 2 0 33 3 3 3( )

0 1 2 3a b a b a b a b a b

3 0 2 1 1 2 0 33! 3! 3! 3!

(3 0)!0! (3 1)!1! (3 2)!2! (3 3)!3!a b a b a b a b

3 2 2 33 3a a b ab b Look! Matches exponent!!

Ex 2) Find the 13th term of the expansion (x + y)15

Look at 2nd variable:

Exponents add to 15

Starts with y0 y1 y2 y3 …So the 13th term would be 1 less: y12

so exponent on x is 15 – 12 = 3: x3

Now the coefficient:

exponent

match exp. on x match exp. on y

3 1215!

3!12!x y 3 12455x y

Ex 3) Find the 7th term in the expansion of (2x – y)10

2nd variable: exponent is 1 less than 7:

1st variable: exponent is 10 – 6 = 4: (2x)4

Coeff:

(-y)6

7th term: 210(2x)4(-y)6 = 210(16x4)(y6) = 3360x4y6

10!

4!6!210

Ex 4) –35a8b3 is what term in (a2 – b)7 ?

Look at 2nd variable

b0 b1 b2 b3 4th term

starts with:

(2y)0 (2y)1 (2y)2 (2y)3 (2y)4

Ex 5) Find the middle term of (x2 + 2y)4

Look at 2nd variable

Exp. on 2nd variable is 1 less than 3:

5 terms middle = 3rd term

Exp. on 1st variable is 4 – 2 = 2: (x2)2

(2y)2

Coeff: 4!

2!2!= 6(x4)(4y2) = 24 x4y2

Homework

#1110 Pg. 542 1 – 23 odd

(Don’t try this on the test!!!)