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9.5 The Binomial Theorem

Homework for section 9.5

P686 #7-11, 27-33, 39 (use Pascal’s triangle, even though it says

to use the Binomial Theorem), 47-51, 55-59, 85, 87

Binomials are polynomials that have two terms.

We will study a formula that gives a quick method of raising a binomial to a power.

0x y

1x y

2x y

3x y

4x y

5x y

1

x y

2 22x xy y

3 2 2 33 3x x y xy y

4 3 2 2 3 44 6 4x x y x y xy y

5 4 3 2 2 3 4 55 10 10 5x x y x y x y xy y

What type of patternsor observations canbe made?

In each expansion, there are n+1 terms.

In each expansion, x and y have symmetrical roles. The powers of x decrease by 1 in successive terms,while the powers of y increase by 1 in successive terms.The sum of the powers of each term is n.

The first term is always xn; the last term is always yn

The coefficients increase and then decrease in asymmetrical pattern.

The hard part is trying to figure out all the coefficients.

That’s where The Binomial Theorem is helpful……

In the expansion of :n

x y

1 1... ...n r rn r

n n n n nx y x nx C nx yy xy y

n r rn rC x y

Where nCr is the coefficient

n- r is the exponent of the x term, and

r is the exponent of the y term.

!! !n r

nC

n r r

It’s also a thingi on yourcalculator…

Sometimes, n rC is written like this:n

r

They both mean the same thing.

7 3C 7!

7 3 !3!

7!4!3!

7 6 53 2 1

7 51

35

7 4C 7!

7 4 !4!

7!3!4!

7 6 53 2 1

7 51

35

8 2C 8!

8 2 !2!

8!6!2!

8 7 6!6!2!

8 72 1

562

28

10

3

10!10 3 !3!

10!7!3!

10 9 8 7!7!3!

10 9 83 2 1

120

7 3C

7 4C

Note how nCr is the same as nCn- r

Another way to find coefficients is by using: Pascal’s

Triangle 0x y

1x y

2x y

3x y

4x y

5x y

6x y

1

x + y

x2 +2xy + y2

x3 + 3x2y + 3xy2 +y3

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

x6 +6x5y+15x4y2+20x3y3+15x2y4+6xy5+y6

x5 +5x4y +10x3y2 +10x2y3 +5xy4 +y5

Pascal’s Triangle

0x y

1x y

2x y

3x y

4x y

5x y

6x y

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 6 15 20 15 6 1

1 5 10 10 5 1

The term for writing out the coefficients of a binomialRaised to a power is: expanding a binomial

or binomial expansion.

Ex: Write the expansion for: 41x

Pascal’s

tells us that the coefficients are: 1, 4, 6, 4, 1

1 4 6 4 1x4 + x3(1)1 + x2(1)2 (1)3 (1)4x+ +

4 4 3 21 4 6 4 1x x x x x

Alternate signs : start with positive first term,then alternate neg/pos..

4 3 24 6 4 1x x x x

Write the expansion for: 42 3x

4 6 4 1(2x)4 + (2x)3(3)1+ (2x)2 (3)2 (3)3 (3)4(2x)+ +1

4 3 216 96 216 216 81x x x x

Finding a term in a Binomial Expansion

Ex:Find the 6th term of the binomial expansion of:

82a b

For 1st term: n=8, r=0 8C0(a)8-0(2b)0

For 2nd term: n=8, r=1 8C1(a)8-1(2b)1

Therefore……

For 6th term: n=8, r=5 8C5(a)8-5(2b)5

56 a3 25 b5

1792a3b5

Find the 9th term of the binomial expansion of:

123 2a b

9th term means that r = ? r = 8

n = ? n = 12

12C8(3a)12-8(2b)8

10264320a4b8

In this example, what is x?

In this example, what is y?

x = 3a

y = 2b(x + y)12

(495)(34)(a4)(28)(b8)

12C8(x)12-8(y)8

Go! Do!