Factoring polynomials with quadratic forms
We're asked to factor 35k squared plus 100k, minus 15. And because we have a non-1 coefficient out here, the best thing to do is probably to factor this by grouping. But before we even do that, let's see if there's a common factor across all of these terms, and maybe we can get a 1 coefficient, out there. If we can't get a 1 coefficient, we'll at least have a lower coefficient here. And if we look at all of these numbers, they all look divisible by 5. In fact their greatest common factor is 5. So let's at least factor out a 5. So this is equal to 5 times-- 35k squared divided by 5 is 7k squared. 100k divided by 5 is 20k. And then negative 15 divided by 5 is negative 3. So we were able to factor out a 5, but we still don't have a 1 coefficient here, so we're still going to have to factor by grouping. But at least the numbers here are smaller so it'll be easier to think about it in terms of finding numbers whose product is equal to 7 times negative 3, and whose sum is equal to 20. So let's think about that. Let's figure out two numbers that if I were to add them, or even better if I were to take their product, I get 7 times negative 3, which is equal to negative 21. And if I were to take their sum, if I add those two numbers, it needs to be equal to 20. Now, once again, because their product is a negative number, that means they have to be of different signs, so when you add numbers of different signs, you could view it as you're taking the difference of the positive versions. So the difference between the positive versions of the number has to be 20. So the number that immediately jumps out is we're probably going to be dealing with 20 and 21, and 1 will be the negative, because we want to get to a positive 20. So let's think about it. So if we think of 20 and negative 1, their product is negative 21. Sorry. If we take 21 and negative 1, their product is negative 21. 21 times negative 1 is negative 21. and if you take their sum, 21 plus negative 1, that is equal to 20. So these two numbers right there fit the bill. Now, let's break up this 20k right here into a 21k and a negative 1k. So let's do that. So let's rewrite the whole thing. We have 5 times 7k squared, and I'm going to break this 20k into a-- let me do it in this color right here-- I'm going to break that 20k into a plus 21k, minus k. Or you could say minus 1k if you want. I'm using those two factors to break it up. And then we finally have the minus 3 right there. Now, the whole point of doing that is so that we can now factor each of the two groups. This could be our first group right here. And so what can we factor out of that group right there? Well, both of these are divisible by 7k, so we can write this as 7k times-- 7k squared divided by 7k, you're just going to have a k left over. And then plus 21k divided by 7k is just going to be a 3. So that factors into that. And then we can look at this group right here. They have a common factor. Well, we can factor out a negative 1 if we like, so this is equal to negative 1 times-- k divided by negative 1 is k. Negative 3 divided by negative 1 is positive 3. And, of course, we have this 5 sitting out there. Now, ignoring that 5 for a second, you see that both of these inside terms have k plus 3 as a factor. So we can factor that out. So let's ignore this 5 for a second. This inside part right here, the stuff that's inside the parentheses, we can factor k plus 3 out, and it becomes k plus 3, times k plus 3, times 7k minus 1. And if this seems a little bizarre to you, distribute the k plus 3 on to this. K plus 3 times 7k is that term, k plus 3 times negative 1 is that term. And, of course, the whole time you have that 5 sitting outside. You have that 5. We don't even have to put parentheses there. 5 times k plus 3, times 7k minus 1. And we factored it, we're done.