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Binomial facts to memorize

A binomial is a kind of polynomial. "Poly" means many, of which the binomial is one, and "bi-" specifies that this particular polynomial has two terms. There are three expression patterns that occur so often when multiplying binomials that it is helpful to simply memorize them by heart. They are the following expressions:

(a+b)^2 = (ab+b)(a+b) = a^2 + 2ab + b^2

(a-b)^2 = (a-b)(a-b) = a^2 - 2ab + b^2

(a+b)(a=b) = a^2 - b^2

It is important to keep in mind that the variables a and b need not refer to integers only, but can also refer to variables. Keep in mind to include the middle term 2ab for the first two of the expressions given to memorize. The reason is this: the presence of the middle term 2ab seems quite counterintuitive, and it must be rigorously memorized. For example, our natural inclination for the first two expressions is going to be to forget about the 2ab and write either a^2 + b^2 or a^2 - b^2, but this is incorrect. It is a common error, but in mathematics, a fatal one, and you will get marked wrong if you write it.