Topic A: Lessons 3-4: Factoring by grouping
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We're asked to factor 4y squared plus 4y, minus 15. And whenever you have an expression like this, where you have a non-one coefficient on the y squared, or on the second degree term-- it could have been an x squared-- the best way to do this is by grouping. And to factor by grouping we need to look for two numbers whose product is equal to 4 times negative 15. So we're looking for two numbers whose product-- let's call those a and b-- is going to be equal to 4 times negative 15, or negative 60. And the sum of those two numbers, a plus b, needs to be equal to this 4 right there. So let's think about all the factors of negative 60, or 60. And we're looking for ones that are essentially 4 apart, because the numbers are going to be of different signs, because their product is negative, so when you take two numbers of different signs and you sum them, you kind of view it as the difference of their absolute values. If that confuses you, don't worry about it. But this tells you that the numbers, since they're going to be of different size, their absolute values are going to be roughly 4 apart. So we could try out things like 5 and 12, 5 and negative 12, because one has to be negative. If you add these two you get negative 7, if you did negative 5 and 12 you'd get positive 7. They're just still too far apart. What if we tried 6 and negative 10? Then you get a negative 4, if you added these two. But we want a positive 4, so let's do negative 6 and 10. Negative 6 plus 10 is positive 4. So those will be our two numbers, negative 6 and positive 10. Now, what we want to do is we want to break up this middle term here. The whole point of figuring out the negative 6 and the 10 is to break up the 4y into a negative 6y and a 10y. So let's do that. So this 4y can be rewritten as negative 6y plus 10y, right? Because if you add those you get 4y. And then the other sides of it, you have your 4y squared, your 4y squared and then you have your minus 15. All I did is expand this into these two numbers as being the coefficients on the y. If you add these, you get the 4y again. Now, this is where the grouping comes in. You group the term. Let me do it in a different color. So if I take these two guys, what can I factor out of those two guys? Well, there's a common factor, it looks like there's a common factor of 2y. So if we factor out 2y, we get 2y times 4y squared, divided by 2y is 2y. And then negative 6y divided by 2y is negative 3. So this group gets factored into 2y times 2y, minus 3. Now, let's look at this other group right here. This was the whole point about breaking it up like this. And in other videos I've explained why this works. Now here, the greatest common factor is a 5. So we can factor out a 5, so this is equal to plus 5 times 10y, divided by 5 is 2y. Negative 15 divided by five is 3. And so we have 2y times 2y minus 3, plus 5 times 2y minus 3. So now you have two terms, and 2y minus 3 is a common factor to both. So let's factor out a 2y minus 3, so this is equal to 2y minus 3, times 2y, times that 2y, plus that 5. There's no magic happening here, all I did is undistribute the 2y minus 3. I factored it out of both of these guys. I took it out of the parentheses. If I distribute it in, you'd get back to this expression. But we're done, we factored it. We factored it into two binomial expressions. 4y squared plus 4y, minus 15 is 2y minus 3, times 2y plus 5.