## Step 3: Factoring binomials

2x

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

^{2}.)

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.

Huh?

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^**= (x

^{2}- 4)^{^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].

x^2 - 4is not a perfect square, it's adifference 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 + 16is aperfect 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 asx^2 - 4,which as I mentioned is adifference 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 + 1Edit: Several times I used the term 'monomial' where I meant 'binomial'.

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

^{2}-4x, to get to the answer 2x(x-2). He did this by expanding the binomial function. The common factor in the equation 2x^{2}-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 2x^{2}-4x. This is because 2x times x = 2x^{2}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

^{2}- 81t^{2}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)