5.4 SAT Math - Factoring

Factoring equations is often referred to as "pulling out numbers."  Your teacher might say, "What can we pull out?"  What that means is taking out numbers that every term has in common in order to make the question simpler.  It's similar to the early work you did with fractions.  When you were simplify fractions, you do something like this: .  You "pulled out" a 5 from top and bottom in order to simplify.   You could have written it as and not actually done the division to simplify.  You still would have done the factoring.

Factoring with variables works the same way, except sometimes you're not going to pull out the same number from top and bottom, you're simply going to pull out whatever it is that all the top numbers have in common, and whatever it is that all the bottom numbers have in common just like the second fraction example.   So let's say you have the expression .  Just looking at the numerator, both terms have a 4 as a common factor, so you can pull that out and be left with .  Looking at the denominator, both terms have a 3 in common, so you can pull that out front too: .  Lo and behold!  What's happened?  You are left with a lovely common factor of 2x + 1  on both top and bottom and you can cross those out, leaving you with the simplified answer of 4/3.  This is why it's not necessary to look for a number that both the top and bottom have in common; more often, it's going to be a variable expression, so just pull out anything you can and see what's leftover.

Factoring doesn't just work with fractions, it works with regular equations, too.  If you see that two terms have a factor in common, it's usually a good idea to factor even if you're not sure that's what the question is testing.  Often, it's the easiest way to solve.  Take a look at this one.

3. If 4a + 4b = 12  what is a + b? 

If you remember that you can factor, you can quickly pull out the common 4 and be left with 4(a + b) = 12, divide by 4 on both sides, and get 3 for your answer.  If you don't remember to factor, it's a much longer, more tedious process.

Factoring with Exponents

Factoring with exponents becomes quite easy if you remember your exponent rules – when set up like a fraction (that is, division), you're subtracting exponents.  That's it.  So something like  quickly cleans up to .  We've seen these before when we covered exponents, but where factoring becomes even more helpful is if you have a complex multiplication problem.  You can not only factor in those circumstances, you can cross-cancel.  Take a look at this one:

First, factor each fraction:

Then cross cancel to eliminate even more:         

And then multiply straight across:
2mp²•3p = 6mp³