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.