The one number that always confuses students in factoring

these are the worst eight problems that students have with factoring and if you are a teacher you are probably shaking your head and saying oh yeah I have seen students make those mistakes plenty times on tests on quizzes and it's usually just a mental error and being a teacher myself I have made the mental air in front of the classroom I probably have even some YouTube videos on them it's a very easy mistake to make so why is it that the sixes give us such a problem well six can be factored down into two different ways right we have 5 * 1 and we have 3 * 2 now a lot of times the confusion comes with the sixes is like when we're doing with multiply with positives and the negative like it could be Nega you know we're dealing with like a negative six but when you're trying to factor something I don't want you to think about the positives and the negatives there's two things that I want you to understand okay there's really two different equations when we're dealing with a is equal to one that we want to focus on there's the one where our last number r c is going to be positive that is going to be looking for a sum when our last number our C our constant is negative that is going to be looking for a difference okay so this is what I want you to mentally understand because I don't think we need to do a long drawn out process to be able to quickly and easily mentally do these now typically the sum is easier for students to do than the difference so let's go ahead and start with that what I want you to do or a quick little tip that sometimes I had my students do when they were first learning factoring is just recognize look at the last number if it's positive let's write the number sum all right so what does sum mean well what sum is going to tell us is that when we're looking at the factors of that term at C we are going to find the sum right we want to add them together so a five I'm sorry why did I write a six that's so weird okay so we have 6 plus 1 is equal to a 7 and a 3 + 2 is equal to a five all right now here's the thing that kind of goes into this which reason I have this plus or minus everything adds into the Plus or the minus right the plus or minus is going to be dealing with your middle term so when you're dealing with a sum and a plus or minus if you're looking for this factored form if you have this as a positive then that's going to mean your factored form let's just do X um let's just do X Plus H that's going to be an X Plus H times a x uh plus I don't know k but if it was negative then that' be an xus an H or an x minus a k so what I'm trying to tell you is when you're dealing with the sum the factors are going to be either both positive or both negative okay but how do you know which one if they're positive or negative look at the middle term if the middle term is positive then you know your two factors going to be positive so let's go over here six sum right so which of my two factors add to five well you can see it's 3 plus 2 is five right since my middle term is positive I know this is going to be a x + 3 * an X+ 2 over here again I have a sum but my middle term is negative that means my two factors both have to be negative well again what two numbers which of my factors add up to give me NE five that's going to be an x - a 3 * an x - A2 let's go and look at the another sum sum positive and positive that means both my factors are going to be positive right and I know 6 and one are going to be my factors that add up to give me a seven so this mean X+ 7 x + 6 * an x + 1 over here my middle term is negative so that's me x - a 6 * an x - won now most students get familiar with the sum and it's the difference that gives us all the trick and T to be honest with you guys that usually is where the mistakes comes into because students a lot of times will say Mr mclogan here the five right I know 3 plus two adds up to five yeah yeah yeah but remember ladies and gentlemen the last term is negative so this is a difference that is the big mistake where students make they don't recognize the difference when the last number is negative your constant is negative you're looking for the difference of your factors so rather than trying to say and it doesn't matter 1 6 - 5 or 1 - 6 right what is the difference of six and five the difference between I'm sorry of six and one is going to be five right so I could say the difference of 6 and 1 is five and I could say same thing 3 plus two it could be positive or negative right because 2 - 3 is negative 1 3 - 2 is 1 so you could say the difference of three and two is one okay so again how do you know if that difference is going to be a positive difference or a negative difference again it's all about the middle term ladies and gentlemen if the middle number is positive then that means you're going to have a positive difference meaning the larger of your factors has to be positive I'll say that again when you have a difference one factor is positive one factor is negative look at your middle term if your middle term is positive that means that the difference is going to be positive right that means the larger of the two factors has to be positive if the middle term is negative that means you have a negative difference that means the larger of your two factors has to be negative so let's go and look at that over here so here I have my two factors right they have a difference of five which I know is going to be six and one since my middle term is negative that means the larger of my two factors right what's larger than six or one six is that has to be the negative over here now it's a positive now the larger of my two factors has to be positive the larger factor between six and one is going to be six cuz I know they have a difference of five so therefore that's an x + 6 * an X - 1 and then over here ladies and gentlemen you can see that the difference here is positive 1 or negative 1 right well which two factors have a difference of one three and two this is positive larger of my two factors has to be positive which is going to be three and then over here the difference is a negative 1 so therefore the larger of the two factors has to be negative so I know this video is not a cure all but hopefully after going through all these examples and maybe doing some practice on your own you will not make this mistake on your own oh