Factoring polynomials by taking common factors
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- [Voiceover] So we're told to factor the polynomial below by its greatest common monomial factor. So what does that mean? So we have these two terms, and I want to figure out their greatest common monomial factor, and then I want to express this with that greatest common monomial factor factored out. So how can we tackle it? Well, one way to start is I can look at just the constant terms, or not the constants, the coefficients, I should say. So I have the 8 and the 12. Then I could say, "Well, what is just the greatest common factor of 8 and 12?" The gcf of 8 and 12. And there are a lot of common factors of 8 and 12. They're both divisible by 1, they're both divisible by 2, they're both divisible by 4. But the greatest of their common factors is going to be 4. So that is equal to 4. So, let me just leave that there. And then we could think about, "What is...?" Well, let me actually write it right over here. I'll put a 4 here. And now we can move on to the powers of x. We have an x squared and we have an x. And we could say, "What is the largest power of x that is divisible into both x squared and x?" Well, that's just going to be x. x squared is clearly divisible by x, and x is clearly divisible by x, but x isn't going to have a larger power of x as a factor. So this is the greatest common monomial factor of x squared and x. Now we do the same thing for the ys. So, we have a y and a y squared. If we think in the same terms the largest power of y that's divisible into both of these is going to be just y to the first power, or y. And so 4xy is the greatest common monomial factor. And to see that we could express each of these terms as a product of 4xy and something else. So this first term right over here, so let me pick a color, so this term right over here we could write as 4xy. That one's actually, that color's hard to see, let me pick a darker color. We could write this right over here as 4xy times what? And I encourage you to pause the video and think about that. Let's see, 4 times what is going to get us to 8? Well, 4 times 2 is going to get us to 8. x times what is going to get us to x squared? Well, x times x is going to get us to x squared. And then y times what is going to get us to y? Well, it's just going to be y. So 4xy times 2x is actually going to give us this first term. So actually just let me rewrite a little bit differently. So it's 4xy times 2x is this first term, and you can verify that. 4 times 2 is going to be equal to 8, x times x is equal to x squared, and then you just have the y. Now let's do the same thing with the second term. And I just want to do this to show you that this is their largest common monomial factor. So the second term, and I'll do this in a slightly different color, do it in blue. I want to write this as the product of 4xy and another monomial. So 4 times what is 12? Well, 4 times 3 is 12. x times what is x? Well, it's just going to be 1, so we don't have to write a times 1 here. And then y times what is y squared? It's going to be y times y is y squared. And you can verify. If you multiply these two you're going to get 12 xy-squared. 4 times 3 is 12, you get your x, and then y times y is y squared. So, so far I've written this exact same expression but I've taken each of those terms and I've factored them into their greatest common monomial factor and then whatever is left over. And now I can factor the 4xy out. I can actually factor it out. So this is going to be equal to, if I factor the 4xys out, you can kind of say I undistribute the 4xy. I factor it out. This is going to be equal to 4xy times 2x plus -- when I factor 4xy from here I get the 3y left over. Plus 3y, and we're done. And you can verify it. If you were to go the other way, if you were to distribute this 4xy and multiply it times 2x, you would get 8 x-squared y. And then when you distribute the 4xy onto the 3y you get the 12xy-squared. And so we're done. This right over here is our answer. The answer is going to be 4xy, which is the greatest common monomial factor, times 2x plus 3y.