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^2−10x+25−36y^2

Accepted Answer

Hello today we're gonna be factoring a polynomial. So the polynomial given to us is x squared minus six. X plus nine minus 16 Y squared. Now, in its current form, this polynomial is not factory bubble. However, if we group up the first three terms together and allow the last term to be its own group, we can factor each of these individual groups. So let's go ahead and factor this. First. Try no meal while the leading coefficient is one. So in order to factor this, we need to go ahead and find two numbers that multiply together to give us nine but add up to give us negative six. While those two numbers are going to be negative three and negative three because negative three times negative three is going to give us positive nine and negative three plus negative three is going to give us negative six. So our first try no meal factors two, x minus three times x minus three. And this could be rewritten as x minus three squared now, 16, y squared can be rewritten as four, Y squared because it is a perfect square. So if we combine our first factor and second factor together, this polynomial could be rewritten as x minus three squared minus four Y squared. Now, if you notice this is two quantities that are perfect squares of each other. So what we have right now is a difference of squares. And what the difference of square states is that if you have a polynomial in the form of a squared minus B squared. This could be factored to a plus B times a minus B. So again, x minus three squared can be rewritten as x minus three times x minus three. So we're going to allow a two equal to the quantity x minus three and four Y squared Could be rewritten as four y times for Y. So we're going to allow be to equal to four Y. Now, what we want to go ahead and do is write this in the form of our difference of squares. So we're going to have A plus B. And A is going to be x minus three plus B, which is four Y times the quantity a minus three, A being x minus three minus B, which is for Y. 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 in understanding how to factor this type of polynomial. And I'll go ahead and see you all in the next video

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