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Factor expression completely. If an expression is prime, so indicate.$5 x+4 y+25 x^{2}-16 y^{2}$...

Question

Factor expression completely. If an expression is prime, so indicate.$5 x+4 y+25 x^{2}-16 y^{2}$

Factor expression completely. If an expression is prime, so indicate. $5 x+4 y+25 x^{2}-16 y^{2}$



Answers

Factor expression completely. If an expression is prime, so indicate. $5 x+4 y+25 x^{2}-16 y^{2}$

Were factoring this expression completely, and I'm going to call this part the first term and this part the second term. So both terms have some common factors that we can factor out. Notice that both terms have why to the fourth in them. So we can factor out y to the fourth. And both terms have y plus two to the third and them. So we config factor out why plus two to the third. So what do we have left from the first term? We factored out the white of the fourth. We factored out the y plus two to the third. So all we have left is one to hold the place of the first term. In the second term we factored out white of the fourth. So we still have a why and we factored out. Why? Plus two to the third. So we still have a white plus two. Now let's simplify that third term. So here we have our white Of the fourth are y plus two cubed and then we can distribute the Y in our third term. So we have one plus. Why squared? Plus two. Why now? Actually, if you rearrange the terms and that factor you get. I'll just start writing the other factors. First, rearrange the terms in that factor. You have one plus two. Why plus y squared and you might realize that that is itself factory ble. So we have white of the fourth we have why plus two cubed. And then when you factor the try no meal, you get one plus y times one plus why you could write that as y plus one if you want instead of one plus Why so all together, then our answer's going to be wide of the fourth times. Why, plus two cubed times y plus one squared.

With the polynomial of four terms, you may be quickly inclined to uh, group used the grouping method, but then getting quickly stuck because there's nothing to greatest common factor out does not mean all is lost. Look at the leading coefficients. We had some likeness going on, so nothing says we cannot reorder these terms. So what if we say this is 16 a squared X plus 16 ace where, why and then we had a negative 25 x minus 25. Why Now we try this greatest common factor again. We can greatest common factor. A 16 a squared out of the first group will be left with experts why we can factor out of negative 25 in the second group left With experts wide moving forward, we now have of the common Roop experts wide to each terms that weaken Greatest common factor X plus y. Don't we left the 16 a squared minus 25. We are still not done. 16. A squared minus 25 can be written. As for a squared minus five squared, it is a difference of squares so it can be expressed. As for a plus five four a minus five along with the experts. Why we accurate out from the start? So in no particular order are factors. Here are experts fly for a plus five for

Two factor this polynomial. It's a try. No meal. So when will? Twice before and adds to negative five negative for in negative one. So we're gonna break part this middle term. Excellent. Fourth minus four. Expert last where minus one Expert life squared. That way we would be able to combine those to get a negative five x squared y squared. That's four wide before you know where you got this expressed with four terms instead of three. And we can do a little reordering before grouping, okay? And we're not gonna do the traditional grouping here because those 1st 3 terms now that we're reordered is a perfect square trying Romeo X squared minus two y squared, squared minus X. Where? Why square, which can be expressed as X y squared. Where are we doing this? Because this is now a difference of perfect squares. So if this is a difference of perfect squares, we have X squared minus two y square plus X y in one factor. And then the other factor X squared minus two y squared plus x one and then the other one minus x y. But I think we're done here, but we're not because our to try no meals each can be factored further. Do a little reordering here expert plus X y minus two last word Samore Reordering entreaties like regular trying told me also a low class negative to and adds to possible and use that break down the middle term weaken right, this is X squared plus two x y minus X y minus two y squared. Similarly, in the second try no meal. What multiplies too negative, Too nasty Negative one negative to in positive Once we break that apart to be X squared minus two x y plus x y minus two. Why, I swear. And now we can do the grouping method with each of these separately motive and kind of simultaneously group the first to get that negative in the second group, we could take out X from the first group and we're left with X plus two why we could take out a negative wine. We're left with experts to why So we have a common group of exposed to why and we took out X minus y. Let's move over the next one we could take out an ex left with X minus two y you could take out why left with ex mice too wide. So our common group is X minus two. Why? When we took out explosive wide and no work done We got this down to all four factors not are None can be factored further. So you no particular order. You need groups of X plus y X minus y exposed to why an X minus two. Why?

Were asked a factor. This expression 25 X squared minus 40 x y and then plus 16 y squared. So what? I know what it's right away is that this is actually five x all squared 25 x squared is and that this is for why all squared And so that leads me to believe that this is a perfect square. In order for it to be a perfect square, it has to be a squared minus to a B plus B squared. So what we imagine is that a Is this a S five X and B is for why so that means that too a B becomes two times five x times for why so that's two times five is 10 times four is 40 x y So we check our middle term up here. Is it 40 x y we say yes, it is. So that means that that is going to factor as a perfect square which just becomes a minus B squared. So in our case, this is gonna become a which is five x minus B, which is four. Why all squared? So that's how we factor that we factor it as a perfect square and it comes out to five x minus four. Why all squared


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