Q:  Multiply: (3x – y)(9x² + 3xy + y²)

A:  To multiply polynomials, we have mostly been taught the word “FOIL”.  Foil stands for “First, Outer, Inner, Last” which is a method of remembering how to multiply the terms.

When I multiply polynomials, I like to think that: Everyone has to touch everyone once, but don’t touch the person you came with 🙂  Right?  When you go to a party, you shake hands with everyone at the party but you don’t need to shake hands with the person you rode with.  What does this mean with math??? Look below:

(3x – y)(9x² + 3xy + y²)

Each blue part needs to multiply each red part, but blue doesn’t mulitply blue (and red doesn’t multiply red):

(3x – y)(9x² + 3xy + y²) = 3x*9x² + 3x*3xy + 3x*y² – y*9x² – y*3xy – y*y²

Now, combine:

= 27x³ + 9x²y + 3xy² -9x²y – 3xy² – y³

Now we can combine like terms, for example: + 9x²y and -9x²y can cancel each other out because the have the same number of x’s and y’s:

= 27x³ + 9x²y + 3xy² –9x²y – 3xy² – y³

= 27x³ – y³