How to Use the Binomial Theorem

The binomial theorem is useful for working out variables to the nth power. For example, to work out (a+b)3 by hand, you would need to use the distributive property on (a+b).(a+b).(a+b), which equals a3+3a2b+3ab2+b3. Figuring out (a+b)10 or (a+b)100 would take you all day to solve. That’s where the binomial theorem comes in handy. The theorem is stated as:

It looks complicated, but it saves a lot of time. Once you’ve worked an example or two, you’ll see how easy it is to actually use.

Sample problem: What is (x + y)4? (Use the binomial theorem).

Step 1: Insert the exponent into the binomial formula, replacing “n”. The exponent in the question is 4, so:

The summation sign means that you are going to sum a series of parts — in this case the parts will be 4…3…2…1…0. If you had a power of 5, the parts would be 5…4…3…2…1…0 and so on.
Step 2: Sum the series of parts. The changes are highlighted in red and blue so you can see them more easily:

This looks complicated by all you are doing is:

  1. reducing the exponent for “x” by 1 in each step
  2. increasing the exponent for “y” by 1 in each step
  3. decreasing the number of items you choose in each step (indicated by the bottom number in parentheses).

Step 3: Use combinatorics to expand the combinations part of the equation. If you aren’t familiar with combinatorics, this combinations calculator walks you through the steps.

Step 4: Evaluate the factorials. This article on factorials will help if you aren’t familiar with factorials.
=x4 + 4x3y + 6b2y2 + 4xy3 + y4
That’s it!

Tip: Note that 0! = 1, and x0 or y0=0.

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