Factoring polynomials - Quadratic forms
To better understand how we can factor second degree expressions like this, I'm going to go through some examples. We'll factor this expression and we'll factor this expression. And hopefully it'll give you a background on how you could generally factor expressions like this. And to think about it, let's think about what happens if I were to multiply x plus something times x plus something else. Well, if I were to multiply this out, what do I get? Well, you're going to get x squared plus ax plus bx, which is the same thing as a plus bx plus a times b. So if you wanted to go from this form, which is what we have in these two examples, back to this, you really just have to think about well, what's our coefficient on our x term, and can I figure out two numbers that when I take their sum, are equal to that coefficient, and what's my constant term, and can I think of two numbers, those same two numbers, that when I take the product equal that constant term? So let's do that over here. If we look at our coefficient on x, can we think of an a plus ab that is equal to that number negative 14? And can we think of the same a and b that if we were to take its product, it would be equal to 40? So what's an a and a b that would work over here? Well, let's think about this a little bit. If I have 4 times 10 is 40, but 4 plus 10 is equal to positive 14. So that wouldn't quite work. What happens if we make them both negative? If we have negative 4 plus negative 10, well that's going to be equal to negative 14. And negative 4 times negative 10 is equal to 40. The fact that this number right over here is positive, this number right over here is positive, tells you that these are going to be the same sign. If this number right over here was negative, then we would have different signs. And so if you have 2 numbers that are going to be the same sign and they add up to a negative number, then that tells you that they're both going to be negative. So just going back to this, we know that a is going to be negative 4, b is equal to negative 10, and we are done factoring it. We can factor this expression as x plus negative 4 times x plus negative 10. Or another way to write that, that's x minus 4 times x minus 10. Now let's do the same thing over here. Can we think of an a plus b that's equal to the coefficient on the x term? Well, the coefficient on the x term here is essentially negative 1 times x. So we could say the coefficient is negative 1. And can we think of an a times b where it's going to be equal to negative 12? Well, let's think about this a little bit. The product of the 2 numbers is negative, so that means that they have different signs. So one will be positive and one will be negative. And so when I add the two together, I get to negative 1. Well, just think about the factors of negative 12. Well, what about if one is 3 and maybe one is negative 4. Well, that seems to work. And you really just have to try these numbers out. If a is 3 plus negative 4, that indeed turns out to be negative 1. And if we have 3 times negative 4, that indeed is equal to negative 12. So that seems to work out. And it's really a matter of trial and error. You could try negative 3 plus 4, but then that wouldn't have worked out over here. You could have tried two and six, but that wouldn't have worked out on this number. Or 2 and negative 6, you wouldn't have gotten the sum to be equal to negative 1. But now that we've figured out what the a and b are, what is this expression factored? Well, it's going to be x plus 3 times x plus negative 4, or we could say x minus 4.