Factoring (Distributive Property in Reverse) Examples

Factor the expression 2x + 4y. Check to see that your answer is correct.

Since each term of the expression has a factor of 2, we can "factor out" a 2 from each term to find that 2x + 4y = 2(x + 2y). To check that our answer is correct, we multiply the factors we found using the distributive property. Do we get 2x + 4y? Thankfully, we do, because otherwise this would be a simply awful example.

What is the greatest common factor in the expression 4xy – 6x²?

First of all, the number part will be the largest number that divides both 4 and -6, which is 2. Now, onto the variables. The variable x occurs once in the first term and twice in the second term (it seems to be gaining momentum!), so we can only use one copy of x. The variable y occurs once in the first term but not at all in the second term, so we can't use any copies of it. Multiplying together our number part (2) and our variable part (x), we can conclude that the greatest common factor is 2x. There...now we can part with our parts.

What is the greatest common factor in the expression 6xy³ – 9x²y³ + 15y²z?

Gee whiz. That's visually a bit daunting, but we just need to take it piece by piece. The number part will be the largest constant that divides 6, 9, and 15, which is 3.

The variable x occurs once in the first term, twice in the second, and not at all in the third, so we don't get to use any copies of x.

The variable y occurs three times in the first term, three times in the second term, and twice in the third term, so we get two copies of y.

The variable z doesn't occur in the first two terms, so we don't get any copies of z. We can conclude that the greatest common factor for this expression is 3y². It's a shame, really...z hardly ever gets any time in the spotlight, and here he is getting dissed again. Chin up, z. Your time will come.

What is the greatest common factor in the expression 3x – xy?

The number part will just be 1, since 3 and -1 don't have any common factors besides 1 and -1. The variable part will be x, so the greatest common factor is just x. Not just x, like x isn't important, we just mean...x isn't really...oh boy. Here we go again.

Factor the expression -8x²z – 16x³. Check your answer.

We could write this expression as 8x²(-z – 2x). But it's kind of messy to have negative signs in front of every term in the parentheses, and we at Shmoop are a bunch of neat freaks. Let's pull the negative sign (factor of -1) out: 

-8x²(z + 2x)

To check the answer, we multiply it out: -8x²(z + 2x) = -8x²z – 16x³. Awesome. That's totally what we were hoping we would get.