Copyright © 2011 Pearson, Inc.

9.2The Binomial

Theorem

Slide 9.2 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Powers of Binomials Pascal’s Triangle The Binomial Theorem Factorial Identities

… and whyThe Binomial Theorem is a marvelous study in combinatorial patterns.

Slide 9.2 - 3 Copyright © 2011 Pearson, Inc.

Powers of Binomials

If you expand (a + b)n for n = 0, 1, 2, 3, 4, and 5, here is what you get:

Slide 9.2 - 4 Copyright © 2011 Pearson, Inc.

Binomial Coefficient

The binomial coefficients that appear in the expansion

of (a +b)n are the values of nCr for r =0,1,2,3,...,n.

A classical notation for nCr , especially in the context

of binomial coefficients, is nr

⎛

⎝⎜⎞

⎠⎟.

Both notations are read "n choose r."

Slide 9.2 - 5 Copyright © 2011 Pearson, Inc.

Example Using nCr to Expand a Binomial

Expand a +b( )4, using a calculator to compute

the binomial coefficients.

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Example Using nCr to Expand a Binomial

Enter 4 nC

r 0,1,2,3,4{ } into the calculator to find the

binomial coefficients for n=4.

The calculator returns the list 1,4,6,4,1{ } .

Using these coefficients, construct the expansion:

a+b( )4=a4 + 4a3b+6a2b2 + 4ab3 +b4 .

Expand a +b( )4, using a calculator to compute

the binomial coefficients.

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Pascal’s Triangle

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Recursion Formula for Pascal’s Triangle

n

r

⎛

⎝⎜⎞

⎠⎟=n−1r−1

⎛

⎝⎜⎞

⎠⎟+n−1r

⎛

⎝⎜⎞

⎠⎟ or, equivalently,

nCr =n−1 Cr−1 +n−1Cr

Slide 9.2 - 9 Copyright © 2011 Pearson, Inc.

The Binomial Theorem

For any positive integer n,

a +b( )n=n0

⎛

⎝⎜⎞

⎠⎟an +

n1

⎛

⎝⎜⎞

⎠⎟an−1b+ ...+

nr

⎛

⎝⎜⎞

⎠⎟an−rbr + ...+

nn

⎛

⎝⎜⎞

⎠⎟bn,

where nr

⎛

⎝⎜⎞

⎠⎟=n Cr =

n!r!(n−r)!

.

Slide 9.2 - 10 Copyright © 2011 Pearson, Inc.

Example Expanding a Binomial

Expand z3 +5x2( )

5.

Slide 9.2 - 11 Copyright © 2011 Pearson, Inc.

Example Expanding a Binomial

We use the Binomial Theorem to expand a +b( )5

where a=x3 and b=−5x2 .

a+b( )5

=50

⎛

⎝⎜⎞

⎠⎟a5 +

51

⎛

⎝⎜⎞

⎠⎟a4b+

52

⎛

⎝⎜⎞

⎠⎟a3b2 +

53

⎛

⎝⎜⎞

⎠⎟a2b3 +

54

⎛

⎝⎜⎞

⎠⎟ab4 +

55

⎛

⎝⎜⎞

⎠⎟b5

=a5 +5a4b+10a3b2 +10a2b3 +5ab4 +b5

Expand z3 +5x2( )

5.

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Example Expanding a Binomial

a =x3 and b=−5x2

a5 +5a4b+10a3b2 +10a2b3 +5ab4 +b5

z3 +5x2( )5= z3( )

5+5 z3( )

4−5x2( ) +10 z3( )

3−5x2( )

2

+10 z3( )2−5x2( )

3+5 z3( ) −5x2( )

4+ −5x2( )

5

=z15 −25z12x2 + 250z9x4 −1250z6x6 +3125z3x8 −3125x10

Expand z3 +5x2( )

5.

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Basic Factorial Identities

( )

( ) ( )For any integer 1, ! 1 !

For any integer 0, 1 ! 1 !

n n n n

n n n n

≥ = −

≥ + = +

Slide 9.2 - 14 Copyright © 2011 Pearson, Inc.

Quick Review

Use the distributive property to expand the binomial.

1. x −y( )2

2. (a+ 2b)2

3. (2c+3d)2

4. (2x−y)2

5. x+ y( )3

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Quick Review Solutions

Use the distributive property to expand the binomial.

1. x −y( )2 x2 −2xy+ y2

2. (a+ 2b)2 a2 + 4ab+ 4b2

3. (2c+3d)2 4c2 +12cd+9d2

4. (2x−y)2 4x2 −4xy+ y2

5. x+ y( )3 x3 +3x2y+3xy2 + y3