A binomial is a polynomial with exactly two terms. Multiplying out a binomial raised to a power is called binomial expansion. Your pre-calculus teacher may ask you to use the binomial theorem to find the coefficients of this expansion.

Expanding many binomials takes a rather extensive application of the distributive property and quite a bit of time. Multiplying two binomials is easy if you use the FOIL method, and multiplying three binomials doesn't take much more effort. Multiplying ten binomials, however, takes long enough that you may end up quitting short of the halfway point. And if you make a mistake somewhere along the line, it snowballs and affects every subsequent step.

Therefore, in the interest of saving bushels of time and energy, here is the binomial theorem. If you need to find the entire expansion for a binomial, this theorem is the greatest thing since sliced bread:

This formula gives you a very abstract view of how to multiply a binomial n times. It's quite hard to read, actually. But this form is the way your textbook shows it to you.

Fortunately, the actual use of this formula is not as hard as it looks. Each

comes from a combination formula and gives you the coefficients for each term (they're sometimes called binomial coefficients).

For example, to find (2y – 1)4, you start off the binomial theorem by replacing a with 2y, b with –1, and n with 4 to get: