The Decomposition Method of Factoring

Hi, I'm Rachel, and today we're going to be going over how to do the decomposition method of factoring. So, the decomposition method is when you have a polynomial where there's a front term, a coefficient before the x-squared. So we need to find out not only the factors of 12, but also the factors of 10 when we're factoring it. Well let's set it up like we're factoring any equation. We know there's going to be two binomials that we're factoring this trinomial into, and we know that the front terms are either gonna be one and 10 or five and two, the factors of 10. We know that the factors of 12 are either going to be one and 12, two and six, or three and four. So we can try out different ways to multiply them together and see what works trying to get a difference of seven. Well, after trying a few out, I notice that multiplying five times three gives us 15, and two times four gives us eight, 15 minus eight is seven. So that's the combination that works. So we want to use the five and the two in the front, and we want to use the three and the four in the back. We want to multiply the five times the three, those are the outside terms, so we're going to put the three there, and then the four times the two, the inside terms, so we'll put the four there. Now, I like to just do two subtraction signs for now, and then we look and see what the signs are going to be. This subtraction sign tells us that there's going to be two different signs. So, we know one's gonna be negative and one's gonna be positive, and there's going to be a difference of seven, negative seven because we see the negative sign there. So, we want a negative 15 and a positive eight to give us negative seven. So the negative is gonna be with the 15 and the plus sign is gonna be with the eight when you multiply those together. And now we have our two binomials, and that is how you factor with the decomposition method. I'm Rachel, and thank you for learning with me today.