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Review of the Methods of Factoring - Problem 2
Factoring an expression with four terms; whenever you see an expression with four terms, that’s a pretty good sign that you’re going to have to factor by grouping. Remember grouping is you’re grouping a pair and a pair and then hopefully when you factor something out those inside terms you’ll be left with are the same, so you can factor out again.
So let’s look at this. The first thing you always want to do is just group up pairs and you want to group up pairs that have something in common. In this case the first two terms have a 2b². The second have nothing in common that’s okay though because we’re going to see that when we factor this out, we’re actually going to be left with an 8 plus 4 as well.
So what we want to do is factor out a common. Group these two together and factor out the common factor. In this case 2b² leaving us with a minus 4. Then we’re left with minus a plus 4 over here. We want to somehow group these together take something out so that this term and the a minus 4 are the same so we have sort of a bigger term here and a bigger term here that both share a factor.
Right now I see these are equal and opposite which means if I take out a -1, these meet with a, a minus 4. You can always check, distribute back in -1 times a is –a. -1 times -4 is +4. So we’re left with 2b², a minus 4, minus 1, a minus 4. Okay so really we have what I would call two complicated terms. We have a term here and a term here and again what we want to do is take out the common factor. They both have a, a minus 4. So we take out the a minus 4, this term is left with a 2b² and this term with a minus 1.
So what we have done is we’ve grouped it up, made each of our insides to be the same, factored that out and we ended up with this. We could almost factor this second term. It’s almost a difference of squares expect that we have these 2 in here. If this 2 wasn’t here, you’d be able to factor it as b plus 1a and b minus 1, but because we have this 2, it’s not quite a square, it’s not going to be able to be factored.
So in order to factor by grouping, group them together, factor it out, factor out again and you’re done.