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The Greatest Common Factor

Been here and done this, but we'll go through a couple of examples to jog your memory. Sorry, we know how much your memory despises physical exercise.

Sample Problem

Factor the polynomial x2 + x.

Since each term has a common factor of x, we can pull out the x and factor the polynomial as the product (x)(x + 1).

Let's make sure this is right. To check, we do the multiplication (x)(x + 1) and look to see that we have the original polynomial back:

(x)(x + 1) = x2 + x

Beautiful. We now have a warm, fuzzy feeling in our chests, and hopefully it has nothing to do with that live gerbil we accidentally ingested earlier.

Sample Problem

Factor the multivariable polynomial 6xy – 3xy2 + 27x2y.

Each term has factors of 3, x, and y. We may need a bigger pair of forceps.

Pulling out 3xy, we can factor the polynomial as (3xy)(2 – y + 9x).

Since the multiple variables make this one a little tricky, make sure we did it right. We'll multiply out the factors we found, and check that the product is the original polynomial. If it isn't, we'll figure out what we did wrong and fix it. Don't GCF (Get Completely Frantic).

(3xy)(2 – y + 9x) = (3xy)(2) + (3xy)(-y) + (3xy)(9x)
  = 6xy – 3xy2 + 27x2y

We did it right. Not that we ever doubted ourselves.

As we go through and learn other factoring techniques, remember this one. Whenever we factor a polynomial, the first thing we want to do is look for a common factor. You can waste your time looking for uncommon factors, but they're rare.

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