Step 3: Factoring binomials

Picture of Factoring binomials
Binomials are expressions with only two terms being added.

2x^2 - 4x is an example of a binomial. (You can say that a negative 4x is being added to 2x2.)

First, factor out the GCF, 2x. You're left with 2x (x - 2). This is as far as this binomial can go. Any binomial in the form 1x +/- n cannot be factored further.

When you have a binomial that is a variable with an even exponent, added to a negative number that has a square root that is a natural number, it's called a perfect square.

x^2 - 4 is an example of this. It can be expressed as the product of the square root of the variable plus the square root of the positive constant, and the square root of the variable minus the square root of the positive constant.


Basically, take the square root of the variable. You'll end up with x. Then square root the 4. You'll end up with 2. If you add them together, you'll get x+2. Subtract them, and you'll get x-2. Multiply the two, and you'll get (x+4)(x-4). You've just factored a perfect square.

If you multiply (x+2)(x-2) together using FOIL, you'll end back up with x^2-4.

(FOIL: First Outer Inner Last, a way of multiplying two binomials together. Multiply the first terms of the binomials (x and x in this case), then the outer two (x and -2), then the inner two (2 and x), then the last terms (2 and -2), then add them all up. x^2 - 2x + 2x - 4 = x^2 - 4.)

This can be done again if one of the binomials is a perfect square, as in this instance:

x^4 - 16 = (x^2 + 4) (x^2 - 4) = (x^2 + 4) (x + 2) (x - 2).

This can be factored further if you bring in irrational numbers, see step [9].

How to factor binomials in the form of (x^3 + b^3):

Just plug into (a - b) (a^2 +ab + b^2). For example, (x^3 + 8) = (x - 2) (x^2 + 2x + 4).

How to factor binomials in the form of (x^3 - b^3):

Plug into (a + b) (a^2 - ab + b2). Note that the first two signs in the expression are switched.

(x^3 - 8) = (x + 2) (x^2 - 2x + 4).

Both examples can be factored further once you learn how to factor trinomials in step [4].
jdevlin826 months ago

x^2 - 4 is not a perfect square, it's a difference of squares. No pun intended, but there's a difference. Take note of the fact that it gets factored out to (x + 2)(x - 2).

Now x^2 - 8x + 16 is a perfect square.
It gets factored out to (x - 4)(x - 4), or just (x - 4)^2. That's what makes it a perfect square. That is not the same as x^2 - 4, which as I mentioned is a difference of squares, because x^2 and 4 are both perfect squares and we are subtracting one from the other.

A perfect square is basically a binomial expression that is a monomial multiplied by itself (squared). It will always result in a trinomial. Even the simplest monomial squared will result in a trinomial. Consider the monomial (x + 1). We see that:

(x + 1)^2 = (x + 1)(x + 1) = x^2 + 2x + 1

Or the negative:

(x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1

Edit: Several times I used the term 'monomial' where I meant 'binomial'.

bseibert3 years ago
Kumon's Math Study Guide provides this question:

Know how to "factor" Binomials:
Problem: -3x^2 + 7x
Answer: 0, 7/3

I thought the obvious answer is pulling out the x for the original binominal which would result in x (-3x+7)

Although I haven't seen this before, I could take each of the factors and have them equal 0 to come up with the answer. Am I missing something?

Example: x = 0 and -3x + 7 = 0
Solve for x, x = 0 and x = 7/3
Phoenixsong (author)  bseibert3 years ago
You are exactly right. The problem should have been worded differently (something like 'Given -3x^2 + 7x = 0, solve for x.').
ilovetine4 years ago
can you please give an example of factoring binomials ?
Mattonator6 years ago
In the binomials section he tries to explain how to factorise 2x2-4x, to get to the answer 2x(x-2). He did this by expanding the binomial function. The common factor in the equation 2x2-4x is 2x, this is because there is a 2x in 2x2 obviously but a 2x inside -4x also. therefore if you times 2x(x-2) out you will get 2x2-4x. This is because 2x times x = 2x2 which is the first part of the equation and 2x times -2 gives the answer -4x because a positive times a negative is a negative number.

Hope this explains it, if not, reply to me
factor, not factorise. you just failed at trying to look smart.
Need more explanation, please. For example, how would you factor 4v squared - 81t squared. I cannot figure it out from the instructable above, or yours. HELP!
Phoenixsong (author)  ladams2345 years ago
4v2 - 81t2

Square root both of the terms in the function. You'll end up with 2v and 9t.

Then, plug it in to the set pattern, (x + y)(x - y).

(2v + 9t)(2v - 9t)
Thanks! I did some research and found the same answer you did. I just wasn't quite understanding it before - thanks again!
Rated 0.5 (worthless). And thats really what this is. You didn't even explain how you got to 2x (x - 2) for the binomials section. This is pretty much what you did: HOW TO FACTOR 1) factor problem 2) get answer.
Nerd rage.