multiplying polynomials

Multiplying polynomials means handing out every term of the first polynomial to every term of the second — a complete round of handshakes, where everyone on one side greets everyone on the other. It is the distributive property applied as many times as needed.

The procedure: multiply each term of the first by each term of the second, using the product rule of exponents (add the exponents when the bases match), then collect like terms to tidy up. For (x + 2)(x + 3) you form x·x, x·3, 2·x, 2·3, giving x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

A useful sanity check: the degree of the product equals the sum of the two degrees. Multiplying a degree-2 by a degree-3 gives a degree-5 result. If your answer's degree does not match, a term was dropped or mis-multiplied somewhere.