Binomial Theorem

How to use the binomial theorem to expand (a + b)n: formula, 2 examples, and their solutions.

Formula

Formula

To expand the power of a binomial (x + y)n:

Write nC0.
Write xn - 0 = xn.
And write y0.

Write +nC1.
Write xn - 1.
And write y1.

Write +nC2.
Write xn - 2.
And write y2.

Repeat this from k = 0 to k = n.

This is the binomial theorem.

Sigma Notation

Meaning of nCkxn - kyk

nCk:
The number of ways
to pick k of y from n trials

Combination

yk: Multiply the picked y factors.
(k of them)

xn - k: Multiply the picked x factors.
(n - k of them)

Example 1

Example

Solution

(x + 4y)3 is the power of a binomial.

So use the binomial theorem
to expand this expression.

Case 1: Pick 0 of +4y.

Write,

the number of ways
to pick 0 of +4y from 3 trials,
3C0

times x3 - 0 = x3

times (4y)0.

Plus...

Case 2: Pick 1 of +4y.

Write,

the number of ways
to pick 1 of +4y from 3 trials,
3C1

times x3 - 1 = x2

times (4y)1.

Plus...

Case 3: Pick 2 of +4y.

Write,

the number of ways
to pick 2 of +4y from 3 trials,
3C2

times x3 - 2 = x1

times (4y)2.

Plus...

Case 4: Pick 3 of +4y.

Write,

the number of ways
to pick 3 of +4y from 3 trials,
3C3

times x3 - 3 = x0

times (4y)3.

So (x + 4y)3
= 3C0x3(4y)0 + 3C1x2(4y)1 + 3C2x1(4y)2 + 3C3x0(4y)3.

3C0 = 1
(4y)0 = 1

Zero Exponent

3C1 = 3/1 = 3
(4y)1 = 4y

3C2 = 3C1
x1 = x
(4y)2 = 42y2 = 16y2

3C3 = 1
x0 = 1
(4y)3 = 43y3 = 64y3

Power of a Product

1⋅x3⋅1 = x3

+3⋅x2⋅4y = +12x2y

+3C1 = +3
x⋅16y2 = 16xy2

+1⋅1⋅64y3 = +64y3

+3⋅16xy2 = +48xy2

So
x3 + 12x2y + 48xy2 + 64y3
is the answer.

Example 2

Example

Solution

The exponent of (a - 2b)8 is 8.

So k goes from 0 to 8.

It says
find the fourth term.

And it says
the first term is a8.

So k is,
start from 0 and count 4:
0, 1, 2, and 3,
3.

So the fourth term is
when k = 3.

So write,

the number of ways
to pick 3 of -2b from 8 trials,
8C3

times a8 - 3 = a5

times (-2b)3.

8C3 = [8⋅7⋅6]/[3⋅2⋅1]

Combination

(-2b)3
= (-2)3⋅b3
= (-8)⋅b3

Cancel the denominator 3⋅2⋅1
and cancel the 6 in the numerator.

Then [8⋅7⋅6]/[3⋅2⋅1] = 8⋅7.

8⋅7⋅(-8)
= 56⋅(-8)
= -448

So -448a5b3 is the answer.