To Factor or to FOIL: Dividing by Zero on the GMAT

by , Aug 17, 2010

We all know not to divide by zero. The reasons for it are pretty straightforward, and for most GMAT students remembering not to divide by zero when solving equations is a no-brainer. Except in this case.

Here is a problem where aspiring GMAT 800's tend to forget that dividing by zero can cause trouble on the Quant section:

If [pmath](x + 4)(3x + 1) = 3x^2 + x[/pmath], what is a possible value of x?

(A) 1

(B) 1/2

(C) 1/3

(D) -1/3

(E) -1/2

There are a couple potential approaches to this problem. We could FOIL the expression on the left side of the equation. That won't take too long, but if we're really up on our game, we might notice that if we factor an x from the expression on the right side of the equation, there will be a (3x + 1) on both sides, which will let us cancel and simplify. That would be faster, and every second helps, so let's use that method.

[pmath](x + 4)(3x + 1) = 3x^2 + x[/pmath]

[pmath](x + 4)(3x + 1) = x(3x + 1)[/pmath]

We cancel the (3x + 1) on both sides, giving us:

x + 4 = x

Now we subtract x from both sides and get:

4 = 0

Wait a minute. Something went wrong. It is quite certain that 4 does not equal 0, so what happened? We can go over our calculations, but we didn't make any errors. And this is the GMAT; 75 minutes are ticking away fast, so we don't have time to ponder the rift in the universe that allows 4 = 0. Let's just do the problem again really quickly with the first method. (We'll take a second look after we finish.)

[pmath](x + 4)(3x + 1) = 3x^2 + x[/pmath]

[pmath]3x^2 + x + 12x + 4 = 3x^2 + x[/pmath]

Now we combine like terms by subtracting [pmath]3x^2[/pmath] and x from both sides, and we solve:

12x + 4 = 0

12x = 4

x = 1/3

So, 1/3 is a possible value of x, and answer choice D is correct.

Now let's take a look at what went wrong when we factored and canceled. When we first pulled out an x, giving us (x + 4)(3x + 1) = x(3x + 1), everything was going fine. We hadn't broken any rules yet.

And then we canceled the (3x + 1). When we cancel in this situation, what we are doing is dividing both sides by (3x + 1). The factors essentially go away, since (3x + 1)/(3x + 1) is always equal to 1. Except when (3x + 1) is equal to zero! Hindsight is 20/20, so let's plug in x = 1/3, and sure enough it turns out that (3x + 1) is zero.

The Takeaway

We can never divide by a variable, or by a variable expression, unless we know that the variable or expression does not equal zero. Remember, canceling is dividing, too.

Keep this in mind and you'll avoid the head-scratching realization that 4=0. This will save you some troubling philosophical pondering, not to mention a lot of valuable time on test day.

Guest post written by Kyle Hausman, one of the GMAT prep experts at Knewton.