In Exercises 103–114, factor completely. x^4−5x^2 y^2+4y^4

Question

In Exercises 103–114, factor completely. x^4−5x^2 y^2+4y^4

Accepted Answer

Hello today we're going to be factoring a polynomial. So the polynomial given to us is x cubed minus seven X squared plus five, X minus 35. Now notice how this polynomial has four terms in total. What we can go ahead and do is factor this by using factor by grouping. And what that means is we're gonna go ahead and group up our first two terms together and our last two terms together. And we want to go ahead and identify a greatest common factor in both of these groups. If we look at our first group we have X cubed and seven X squared. Both of these terms have a common factor of X squared. So the first group is going to have the greatest common factor of X squared. Now in our second group we have five X and negative 35. So both of these factors are divisible by five. So the second group has the greatest common factor of five. If we factor out the greatest common factor from both of these groups, the first group is going to give us x squared times the quantity x minus seven. And if we factor out five from the second group we're gonna get positive five times the quantity X -7. Notice how both of these quantities have the common term x minus seven. So we can go ahead and combine both of these quantities together by just adding the leading coefficients together. Doing so is going to give us x squared plus five times X minus seven and x squared plus five is not factory bubble. So 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 D. So I hope this video helps you in understanding how to factor by grouping, and I'll go ahead and see you all in the next video.

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