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Factoring the Sum or Difference of Cubes - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

When factoring trinomials, we can learn certain patterns of factoring the sum or difference of cubes. When factoring sum of cubes expressions, we will always end up with the binomial (a + b) multiplied by the trinomial (a^2 - ab + b^2). When factoring the difference of cubes, we will always end up with the binomial (a - b) multiplied by the trinomial (a^2 + ab + b^2).

Factoring trinomials. So there's some things that we know how to factor right away, okay? Just by time and experience and just dealing with them a lot. Behind me I have 2 examples. x squared plus xy plus y squared. Hopefully, you look at that and you recognize, Oh. That's just going to be x plus y squared. Likewise, x squared minus y squared, hopefully you recognize that to be x+y times x-y. Okay. Those are 2 that we know.
We can also deal with the what's called the difference or the sum of cubes. And what I mean by that is if we are looking at a cubed plus b cubed. We can factor this as well and how we do that is a formula similar similar to this. Okay, just in terms of having to memorize a new formula. And how this works is we have a binomial a+b and then we also have a trinomoal which is going to be a squared minus ab plus b squared. Okay? So just like these are formulas, this is a new formula as well. Okay? This will also work for the so I will call this the sum of cubes because we are adding 2 cubes together. This will also work for the difference of cubes, a cubed minus b cubed. And how this works is almost identical formula except instead of dealing with a+b, we now deal with the minus and then it's a squared plus ab plus b squared.
Now it can be fairly, these similar these formula do look quite similar so it can be kind of hard to distinguish them. The way I remember, and if you have another way that's fine as well, is the sign of these first two binomials, the 2 terms one is going to be the same as the original sign. So here we're dealing with addition, here we're dealing with addition, subtraction, subtraction. That's why we erase my parentheses. Okay.
We then have the a squared. All the values are the same. Okay, but then the second sign is going to be the opposite. So here we started with plus, this goes the minus. Here we started with the minus, this goes to plus. Okay, so if the second sign is oppoite and the third is always going to be positive.
So as long as you can remember the ab is squared ab and b squared, just remember that your signs are the same as you started, opposite and then always positive. There just 2 more formulas we need to add to a repertoire in trifactoring.

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