First, try these practice problems.

1.

(A)

(B)

(C)

(D)

(E)

1.

(A)

(B)

(C)

(D)

(E)

3. If

(A)

(B)

(C)

(D)

(E)

4. If

(A)

(B)

(C)

(D)

(E)

If these are easy for you, you probably have already mastered factoring: kudos to you! If these confuse you, you have found just the post you need.

You don’t need to know any of this terminology for the QUANT, but we need it just to talk about these ideas in words.

A **binomial** is a polynomial with two terms: all five answer choices to question #1 are the product of two binomials. A **quadratic** is a polynomial with three terms whose highest power is x-squared: the stem of question #1 is a quadratic, and the stem of question #2 is a ratio of two quadratics. To factor a quadratic is to express it as the product of two binomials. Question #1 is a straightforward “factor the quadratic” problem. Question #2 involves factoring both quadratics, in the numerator and in the denominator, and then cancelling a common factor.

Technically, any quadratic could be factored, but often the result would be two binomials with horribly ugly numbers — radicals, or even non-real numbers. You will **not** have to deal with those cases on the QUANT. We call a quadratic “factorable” if, when you factor it, the resulting equation has only integers appearing. You will only have to factor “factorable” quadratics on the QUANT.

Sometimes, factoring quadratics involves quadratic like that in #3, with a leading coefficient (the coefficient of the

The best way to understand factoring well is first to understand FOILing well. Suppose we multiply two binomials:

Notice, if we follow the FOIL process forward, then we see two things. First, the middle coefficient, the coefficient of the **If I want to factor any polynomial of the form \bold{x^2 + bx + c}, then to factor it, we are looking for two numbers that have a product of c and a sum of b**.

Things get a little complicated when some negative signs are floating around, so here’s a table, for all cases, for the two numbers,

BTW, I highly recommend NOT memorizing the above chart, but rather, thinking it through, and doing FOIL examples for each case, to convince yourself of the patterns and to ingrain them into your memory.

So, for example, suppose we want to factor: **difference** of

Voila! Having read this, see if you now can figure out questions #1 & #2 above

Most QUANT test takers will not see this topic. Only if you anticipating getting the vast majority of questions on the Quant section correct should you even read this section.

Suppose you have to factor something like

(i)

(ii)

(iii)

This is not quite as methodical and left-brain as factoring in the easy case above. This involves a certain amount of number sense and a certain amount of pattern matching. For

If that makes sense, and you feel up to the challenge, try #3.

Factoring quadratics is such a widely used skill in algebra that you are likely to see something such as Question #1 or #2 on your QUANT. Here’s another practice question, with its own video explanation.

1.

We need two numbers,

That’s the one we want! The pair we want is

**Answer = E**

2. For this one, we need to factor both the numerator and denominator. In the numerator, we have

Now, the denominator,

We see that we have a common factor of

**Answer = B**

3. This is the hard one, definitely a 168+ level question. We need numbers

This means that

Thus, we see:

**Answer = B**

4. Our first step is to simplify the first equation, which we do by multiplying both sides by

Now we will use the FOIL method to multiply the left side of the equation together. This gives us:

So now we have

This is the exact expression we were asked to evaluate.

**Answer = C**