Factoring: Special Cases Part II - Concept
Solving rational equations is substantially easier with like denominators. When solving rational equations, first multiply every term in the equation by the common denominator so the equation is "cleared" of fractions. Next, use an appropriate technique for solving for the variable.
One type of factoring you guys will be practicing is factoring by grouping. Factoring by grouping is used mostly when you have polynomials that have four terms and we'll see examples in a second. The four terms often means your highest exponent number is 3 or it's a third degree polynomial, I'll show you an example once we get there. Also useful factoring by grouping is useful for polynomials with 4 terms. What you want to look for is the greatest common factor binomial, usually what we're used to finding is that the greatest common factor is a monomial, we're looking for a binomial and here's an example to show you that.
This is a problem that's like halfway done, so sometimes your homework problems start out like this they start like in the middle of what we would call "factoring by grouping problems." If you look at this term, it's already broken into its factors and this term is also broken into its factors. This represents two different products and if you look at each of these products you'll notice what they have in common. They both have that y+2 term so what I can do to write the factored version of these two things, I would write y+2 times y squared plus 8. That's the factored form of this original sum of two products. It's tricky because it's all kinds of vocabulary so I want to say that one more time, this represents the factored form of this sum, sum meaning addition of two different products.
Okay, the trick is that often times your homework problems aren't going to look like this, they're not going to be a sum of two different products. What they're going to be giving you is this 4 term polynomial, so I'm going to like backup in this problem, this step, your homework isn't going to give you this. What your homework is going to give you is this thing multiplied out into 4 different terms it looks like this; y to the third plus 2y squared plus 8y plus 16, you'll notice what I did was just distributing, so when you come to your homework you're going to be starting here it's going to give you that 4 term polynomial where the degree is 3 the highest exponent is 3.
Here's what you guys are going to do when you get to your homework problems, first thing you're going to do is make sure this is in standard form, standard form meaning the exponents are decreasing, and in my case, this problem they are.
Next thing I want to do is cut this polynomial in half and then I'm going to be factoring each side all by itself. Like the first thing I did before I showed you this video was I factored y to the third plus 2y squared and that's how I got this piece, the common monomial there was y squared, that was the result. Then I factored this half, I looked for what multiplies to 8 and also into 16 and that was this 8 common factor, there is my resulting binomial when I like undistribute.
That's how I got from step 1 to step 2 to step 3, this is how you guys are going to be doing your homework. I know this is a little tricky because I like started in the middle and then went backwards. I promise it'll make more sense once you get going on your homework problems just remember to look for your polynomial in standard form cut it in half and then factor each half on it's own, once that's done, look for the common binomial and then write your answer as the final product.