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Video transcript

we're asked to factor for y squared plus 4y minus 15 and whenever you have an expression like this where you have a non 1 coefficient on the y squared around 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 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 needs to be equal to 4 so let's think 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 the product is negative so when you take two numbers of different signs and you sum them you're kind of view it as the difference of their absolute values so if that confuses you don't worry about it but this tells you that the number since they're going to be a 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 there's just still too far apart what if we tried what if we tried 6 and negative 10 then you get a negative 4 if you added these two we want to 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 @n y so let's do that so this 4y can be rewritten as negative 6y plus 10y all right because if you add those you get 4y and then the other sides of it you have your 4y squared 4y squared and then you have your -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 terms so let's see this we 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 two Y so if we factor out two Y we get 2y times 4y squared divided by 2 y is 2y and the negative 6y divided by 2y is negative three negative three so that gets fact 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 the greatest common factor is a 5 so we can factor out a 5 so this is equal to plus 5 times 10 Y divided by 5 is 2 y negative 15 divided by 5 is 3 and so we have 2y times 2y minus 3 plus 5 times 2y minus 3 so now you have two terms so now you have two terms and 2y minus 3 is a common factor to both so let's factor out a 2 y minus 3 so this is equal to 2y minus 3 times 2y times that 2y plus plus that 5 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 and 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 2 y -3 times 2y plus 5