# Binomial theorem

The binomial theorem is used to expand polynomials of the form (x + y)^{n} into a sum of terms of the form ax^{b}y^{c}, where a is a positive integer coefficient and b and c are non-negative integers that sum to n. It is useful for expanding binomials raised to larger powers without having to repeatedly multiply binomials. The binomial theorem is written as:

## Breaking down the binomial theorem

If you aren't familiar with the symbols in the binomial theorem, it may seem fairly daunting, but once you break it down piece by piece, the binomial theorem is relatively simple to use. We should be fairly comfortable with exponents, but refer to the exponents page for a refresher if necessary.

### Summation

The symbol, , is the capital Greek letter Sigma. In math, it is referred to as the summation symbol. Along with the index of summation, k (i is also used), the lower bound of summation, m, the upper bound of summation, n, and an expression a_{k}, it tells us how to sum:

Example

Evaluate :

### Combinations

The in the binomial theorem is a combination (specifically a combination without repetition) that is referred to as the binomial coefficient. It is read as "n choose k." Briefly, n choose k indicates how many possible ways there are to choose k elements from a set of n. We will not go into combinations in depth at this point, but will provide a formula that will allow you to use the binomial theorem.

In the formula, the "!" indicates a factorial. The factorial of an integer is the product of all positive integers (0 is not included) that are equal to or less than the integer.

Example

Evaluate 5!:

5! = 5 × 4 × 3 × 2 × 1 = 120

### Using the binomial theorem

Though the examples above use relatively simple examples, the concepts are all the same when using the binomial theorem, and we evaluate the binomial theorem by plugging all the appropriate values into the formula.

Example

Expand (x + y)^{3} using the binomial theorem:

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