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Review of the Methods of Factoring - Problem 4
Factoring a statement with only two terms; so there’s really only a few things we look for when we’re trying to factor a statement with two terms. They’re either going to be the difference of squares, or the sum or difference of cubes.
So looking at what I have behind me I have 8x³ minus 27. I know that 27 isn’t 8² so therefore I can rule out the difference of squares and I’m only left with sum or difference of cubes. Obviously we have a minus sign so obviously the only chance that this could be is the difference of cubes.
So the difference of cubes formula, hopefully you can remember is a³ minus b³ will factor to a minus b, the binomial of this sign agrees and then we’re left with a² plus the opposite sign ab and note there is no 2 here, it’s not like the one we’ were Foiling out a square where we have a 2 in there, just ab and then always plus b² at the end.
So really all we have to do for this problem is to figure out what our a is and what our b is. 27 is what cubed? 3, that’s pretty straight forward, so then we know that we are left with -3, +3, +9 and then all we have to do is to figure out what our a³ is. What cubed will give us 8x³? Obviously it has to be a single x and 8 is 2³, so therefore our a is just going to be 2x. Plug in your 2x in. 2x, 2x quantity squared is going to be 4x² and then this is a 2x in here and then plus 9.
This can be simplified to be 6x everything else still remains the same, so instead of rewriting everything, just imagine this is a 6x, and what we’ve done is factor the difference of cubes.