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Factoring Out the Greatest Common Factor

In factoring out the greatest common factor, we must first find the greatest common factor. So what we’re going to do is we’re going to write down each term of this expression. So we have 12x cubed; then write that down as its prime factorization, we’ve got 2 times 2 times 3 then times three x’s. Okay so next is the 16x squared. So we’ve got 2 times 2 times 2 times 2 times x times x. Next we’ve got negative 8x. Even though negative 1 is not its prime factor, we can still write it as a factor. So we’ve got negative 1 times 2 times 2 times 2 times x. So now, let’s circle what they all three have in common. They all have a 2 in common. They also have another 2 in common. And it looks like they have an x in common. We don't see anything else that all three of them have in common. So, I know that my greatest common factor is going to consist of 2 times 2 times x, which when we multiply all this together, we simply get 4x. So, out of this expression, we have to factor out this 4x. So here's what we are going to do: we automatically write this 4x on the outside, and then what we’re going to do is we’re going to put parenthesis around what needs to be on the inside. So notice, with this 4x, the things that are left over from this term, 12x cubed, are the 3, and the other two x's. So we have to write that down. So we have 3 (and if we multiply them together we have x squared). Now, notice there is that plus sign right there. So, off course we are going to write down the plus sign. So now moving along to the 16x squared, we've got a 2, another 2, and an x left over. So, 2 times 2 is 4 and then there's that x. Now, the only thing that we’re missing from the negative 8x would be the negative 1 and the 2. So, negative 1 times two is simply negative 2. So, I order to check ourselves, remember we can always use the distributive property by multiplying everything back out. Okay. So just quickly checking ourselves 4x times 3x squared gives us this 12x cubed, so that works out. The 4x times the 4x gives us our 16x squared, so that checks out. Then the 4x times the negative 2 gives us the negative 8x. So, this expression factored is just this answer down here.

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