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in the last video we looked at three different examples really is a bit of a review of some of our factoring techniques and also to appreciate when we might want to apply them and we saw in the first example that it was just a process of recognizing a common factor and once we factored that out we were done in the second example there was a common factor for but then after that we used you could say our most basic factoring technique or one of our more basic factoring techniques where we say okay what two numbers add up to this middle to the first-degree coefficient and then their product is the constant and we were able to factor the expression and then in the third example we once again started off by factoring out a common a common value which in this case was 30 and we could have done it the same way we did the second one or we could have immediately recognized that this is a perfect square polynomial but either way we were able to factor the expression let's keep going to see if we can tackle some other types of polynomials that might require some other techniques so let's say we have the expression 7x squared minus 63 like always pause this video and see if you can factor that all right well I've intentionally designed all of these so that you can check whether there's a common factor across the terms and here they're all divisible by seven so if you factor out a seven you're going to get seven times x squared minus nine now you might immediately recognize this as a difference of squares you have x squared minus this right over here is three squared minus three squared and so if the term difference of squares or how to factor them is completely foreign to you I encourage you to watch the videos on factoring difference of squares or do a search on Khan Academy for difference of squares but you will see when you have a difference of squares like that it can be factored as seven this is just a seven out front and then this part right over here in a different color this part right over here can be written as X plus three times X minus three it is x squared minus three minus two three squared now one thing to appreciate this really isn't a different technique than the one that we saw in the previous video if we just focused on x squared minus nine you could use this as x squared plus zero X minus nine in that case you'd say okay what two numbers get me a product of negative nine and add up to zero well if I need to get a product of negative nine that must that means that they must be different signs positive and negative otherwise they're the same sign you get a positive here so they're different signs and nine only has three factors one so you could either have one and nine or you can just like two combinations here you could either have one or nine and three and three and if you make one negative or nine negative that's not going to add up to zero but if you make one of these threes negative that does add up to zero so you say okay well my two numbers are going to be negative three and three and so it's going to be X minus 3 times X plus 3 and once again I'm just focusing on what was inside the parentheses right over here you'd put that 7 out front if we're doing this exact same expression but if you recognize it as a difference of squares it might happen for you a little bit faster let's do one more example so let's say that I have 2x squared plus 7x plus 3 so in general when my coefficient on the second degree term here is not a 1 I try to see if is there a common factor here but 7 is indivisible by 2 and neither is 3 so I can't use the techniques that I used in the last few videos or even over here where I say oh there's there's a common factor and get a leading coefficient of 1 so if you see a situation like that it's a clue that factoring by grouping might apply here and factoring by grouping and on some level everything that we've just done now you could view special cases of factoring by grouping but factoring by grouping you say okay can I think of two numbers that add up to this coefficient so a plus B is equal to 7 and a times B instead of just saying it needs to equal two three it actually needs to be equal to three times this three times the leading coefficient the coefficient on the x squared term so it needs to be equal to three times two if you think about it we've always been doing that but you're the other examples we gave the leading coefficient wasn't one so when you know where you took the constant term and multiplied it by one you're gonna say overall a times B needs to be equal to that constant term but if you want to talk about it more generally it should be a times B should be the constant term times the leading coefficient and then the introduction to factoring by grouping we explain why that works you should never just accept this as some magic formula it makes sense for a very good mathematical reason but once you accept that then it's useful to be able to apply the technique so can we think of two numbers that add up to seven and whose product is equal to six and they're going to have to be the same signs it's this is a positive value and they're going to be positive because they're adding up to the same side as they're adding up to a positive value their goal is going to be positive well let's see one and six seems to work 1 plus 6 is 7 1 times 6 is 6 so in factoring by grouping we write rewrite our expression where we break this up between the a and the B so I can rewrite this as 2x squared plus 6x plus I could write 1 X actually let me just do that plus 1 X plus 3 and as you can see the 7x is a different color the 7x has been broken up into the 6 X + 1 X so the whole exercise I just did is to see how we can break up this first degree term right over here but then what's useful about this is now we can essentially apply the just we can undistribute the we can do the reverse of the distributive property twice so for these first two terms is in a different color than I just use these first two terms you see a common factor 2x squared + 6 X they're both divisible by 2 X so let's factor out a 2x out of those first two terms so if you do that 2x squared divided by 2x you're just going to have an X left over + 6 X divided by 2 actually just going to a three and then you have plus and then over here this is a special situation where X plus three there is no common factor between X and three so we'll just rewrite that X plus three but when I put a parenthesis on it which is equivalent to writing it without a parenthesis you might see something else well I can undistribute or I can factor out an X plus three so what happens if I do that I'm going to get an X plus three and then I'm going to have left over in this term if i factor it out an X plus three I'm just going to have a 2x left over 2x and then this term if i factor out an x plus three well I'm just gonna have a one left over plus one let me do in that same color having trouble switching colors today 2x plus one and we are done so as I said these are all various techniques on some level factoring by grouping is sometimes used as the hardest one but I say hard in put parentheses because everything we did is just a variation really a special case of factoring by grouping and as you can see it's all about well two numbers that add up to that middle coefficient down the first degree term when it's written in standard form and their product is equal to the product of the constant and the leading coefficient and if you do that you break it up it works out quite nicely where you keep factoring out terms and this one on some levels is a little bit more subtle because you have to recognize that this X plus 3 has a 1 coefficient out that on there implicitly 1 times X plus 3 is the same thing as X plus 3 and then see that you can factor out an X plus 3 out of both of these terms and then what you do that you're going to be left with the 2x plus 1 but all of these if you really feel comfortable with this arsenal of techniques you're going to be pretty you're going to do pretty well and frankly if none of these work well you might already be familiar with the quadratic formula or you might be soon to learn it but that's when the quadratic formula might be effective is if none of these techniques work