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“y in terms of x”: When can I factor by grouping?

Question

When can I factor by grouping? According to the video explanation: "You can factor by grouping if there are 4 terms and there exists some sort of pattern between the terms." Is this the only case where this can be done? Are there additional videos you can point me to that explain this a little more clearly?

Answer

Now, factoring by grouping is a special case. It will not appear often and, if it appears, I think it will be pretty obvious when you need to use it because you will have four terms—no two terms that you can combine by addition or subtraction. And you should be able to figure out pretty quickly whether there is a pattern.

Usually, when we have two things like (2x + 1)(x - 3) and we multiply using FOIL, we get four terms and we can combine two of them (in this cases the terms ending in "x" - not x^2) so we have three terms. When we factor by FOIL, we are reversing that.

But when we multiply two things like (2x + 1)(x - 2y), we will get four unlike terms: 2x^2 + x - 4xy - 2y. We can't combine any of these terms -- we have one x^2 term, one x term, one xy term, and one y term. They all different.

So when we have the reverse situationa polynomial with four unlike termsand we want to factor:

2x^2 + x - 4xy - 2y

I think "okay, I have four terms and I can't add or subtract any of these together, so let me check whether I can factor by grouping."

The key is that I can group the four terms into groups of two and and use GCF (greatest common factor) factoring so that each of the two groups have one factor that is the same. That's what the video meant by "pattern":

(2x^2 + x) - (4xy - 2y) =

x(2x+1) - 2y(2x+1) =

(x - 2y)(2x +1)

Another example:

20 - 3xy - 20y + 3x

I see four unlike terms and it looks like I can do some factoring. Let me try grouping the 20s and the 3s together:

(20 - 20y) + (3x - 3xy) =

20(1-y) + 3x(1-y) =

(20 + 3x)(1 - y)

So i've factored by grouping.

You can check out this video from Khan Academy:

Factor by grouping and factoring completely

Khan Academy is a good resource for foundation skills.

Also, if you'd like a more in-depth review of basic concepts, I'd recommend McGraw-Hill's Conquering the New GRE Math. You can read our review here.

Link

http://gre.magoosh.com/questions/145

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