In Exercises 65–92, factor completely, or state that the polynomi...

Question

In Exercises 65–92, factor completely, or state that the polynomial is prime. x^2 y−16y+32−2x^2

Accepted Answer

Hello today we're gonna be factoring a polynomial. So the polynomial given to us is X cubed plus three, X squared -16, X -48. So if you notice here we have a polynomial of four terms. Whenever you see a polynomial for four terms, you want to go ahead and try factoring by grouping. What that means is we're gonna go ahead and group up the terms in groups of two. So we're gonna group up our first two terms together and our last two terms together. And what we want to go ahead and do is identify the greatest common factor from both of these quantities. If we take a look at our first quantity X cubed plus three X squared, notice how both of these terms have a common factor of X squared. So the greatest common factor for our first group is going to be X squared. Now let's take a look at our second group in our second group, both of the terms have factors of 16 because we have negative 16 and negative 48. What we want to do though is go ahead and factor out a quantity that turns the first term positive. So what we want to do is factor out negative 16. So the second group is going to have the greatest common factor of negative 16. Factoring out the greatest common factor from the first group is going to give us X squared times the quantity X plus three. And factoring out negative 16 from the second group is going to give us negative 16 times the quantity X plus three. Now notice how both of these terms have a common factor of x plus three. That makes these two terms combine nable. So what we want to go ahead and do is combine everything here into a single term and we can do that by just adding the leading coefficients together. Doing so is going to give us the quantity X squared minus 16 times X plus three. Okay, now we have everything in one term but this is not fully factored because our first quantity here is a difference of squares. What the difference of square states is that if you have a polynomial in the form of a squared minus B squared, this can be factored to a plus B times a minus B. So A and B are just going to be the perfect squares or the factors of the perfect squares. So for our first term here X square could be rewritten as x times X. So A is going to equal to X and 16 can be rewritten as four times four so B is going to equal to four. Now, what we want to go ahead and do is plug in our A and B into the difference of squares. Doing so is going to give us a plus B or X plus four times a minus B or x minus four. And keep in mind that both of these quantities are still a product of with this original factor. So this is going to be X plus four times X minus four times X plus three. And this is going to be the fully factored version of our given polynomial. And that means that the answer to this problem is going to be a So I hope this video helps you understanding how to factor by grouping and I'll go ahead and see you all in the next video.

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