Factor the polynomial.x2−13x+40x^2-13x+40x2−13x+40

(x−5)(x−8)\left(x-5\right)\left(x-8\right)(x−5)(x−8)

Factor the polynomial. x 2 − 13 x + 40 x^2-13x+40 x 2 − 13 x + 40

Alright, everyone. Let's just jump right into this practice problem here. We have a polynomial, x squared minus 13x plus 40. So if you'll notice here, I don't have a greatest common factor. I can also can't group this because I don't have 4 terms. And if you'll notice, this 40 here isn't a perfect square or a perfect cube, so we definitely can't use our special product formulas. And whenever this happens and we have an expression in this form that's in it's in standard form, we're gonna use the AC method. Alright. So we're gonna use the ac method. In this case, what happens is my a is just equal to 1. My b term is equal to negative 13, and then my c term is equal to 40. Remember what the first sort of step is in solving these types of problems? Build out this little table over here, And what happens is you want a times c. You want two numbers that factor to a times c. So in other words, I need 2 factors that will multiply to 40. And those factors need to add to the b term, which in this case is negative 13. In this case, you always have to use you always have to use the sign that's before that term. So b is not 13, b is negative 13. Okay? So be really careful about that. So let's get started here. Just start at 1 and then just keep going up from there. So one times 40 is 40, but also negative one times 40 is is negative or negative one times negative 40 is 40. What about 2? 220 makes 40. Negative 2 and negative 20 also make 40. Let's keep going. Does 3 work? Well, 3 won't work because 3 doesn't multiply by anything to get to 40. What about 4? 4 times 10 works, so negative 4 times negative 10 also works. What about 5? 58 work as well. So I've got 5 and 8, then 5 and negative 8. And then what happens is 6 doesn't work and 7 doesn't work either. And if I use eights, I'm really just going backwards again. So this is actually all of my unique factors over here. So let's just go ahead and add this stuff up. This is just gonna be 41. Doesn't work. Negative 41. Doesn't work. 2 to 20 makes 22. Doesn't work. Negative 2 make this just makes negative 22. That doesn't work. 4 and 10 makes 14, doesn't work. Negative 14 doesn't work either. Very close. What about 5 and 8? 58 make 13, which is close. We want 13, we want negative 13. So that doesn't work. So, it turns out it's actually just gonna be these two numbers because they add to negative 13. So these are our two numbers. This is my p, and this is my q over here. And so basically, what I found here is I found the two numbers that multiply to 40 and that add to negative 13, and both of these things are negative. So remember how this works out plus negative 5, so this is actually x minus 5, and this becomes x minus 8 over here. If you reverse the order, it wouldn't matter. It's still the same exact expression. Notice how if you foil this out, what happens here is you're gonna get the x squared, the negative 8, and the negative 5 will combine to 13, and the negative 5 and negative 8 will multiply to 40. Alright? So that's all you that's all there is to it. List out your factors, figure out which one of them works, and you're basically done. So that's the answer. Let me know if you have any questions. Thanks.