Factoring polynomials with quadratic forms
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Let's see if we can use our existing factoring skills to factor 30x squared plus 11xy plus y squared. And I encourage you to pause the video and see if you can handle it yourself. Now, the first hint I will give you-- and this might open up what's going on here-- is to maybe rearrange this a little bit. We could rewrite this as y squared plus 11xy plus 30x squared. And my whole motivation for doing that-- there are ways to factor a quadratic where your first coefficient, your coefficient on this first term, is something other than 1. But we haven't seen that yet. And so rearranging it this way, this got us a little bit more into our comfort zone. Now our coefficient is a 1 on the y squared term. So now we can start to think of this in the same form that we've looked at some of the other factoring problems. Can we think of two numbers whose product is 30x squared and whose sum is 11x? Notice, 11x is the coefficient on y. We have y squared, some coefficient on y. And then in terms of y, this isn't in any way dependent on y. So one way to think about this, if you knew what x was, then this would be a quadratic in terms of y. And that's how we're really thinking about it here. So can we find two numbers whose product is 30x squared and two numbers whose sum is the coefficient on this y term right here, whose sum is 11x? So let's just think about all of the different possibilities. If we were just thinking about two numbers whose product was 30 and whose sum was 11, we would be thinking of 5 and 6. 5 times 6 is 30. 5 plus 6 is 11. It's some trial and error. You could have tried 3 and 10. Well, that would have been-- 13 would be their sum. You could have tried 2 and 15. That wouldn't have worked. But 5 and 6 does work here, so we've already seen that multiple times. So 5 and 6 would work for 30, but we have 30x squared. So what if we have 5x and 6x? Well, 5x times 6x is 30x squared, and 5x plus 6x is 11x. So this actually works. So then our factoring or our factorization of this expression is just going to be y plus 5x times y plus 6x. And I'll leave it up to you to verify that this does indeed, when you multiply it out, equal this up here.