When multiplying, things are put together. When factoring, things are pulled apart. Factoring is a critical skill in simplifying functions and solving equations.

There are four basic types of factoring. In each case, it is beneficial to start by showing a multiplication problem, and then show how to use factoring to reverse the results of that multiplication.

"Pulling Out" Common Factors

This type of factoring is based on the distributive property, which states: 

$2x(4x^2-7x+3)=8x^3-14x^2+6x$

When factoring, this property is done in reverse. Therefore, starting with an expression such as the one above, it can be noted that every one of those terms is divisible by $2$. Also, every one of those terms is divisible by $x$. Henc, one can "factor out," or "pull out," $2x$.

$8x^3-14x^2+6x=2x(? { } - ? { } + ? { } ).$

We now divide each term with this common factor to fill in the blanks. For instance, $8x^3$divided by $2x$ equals $4x^2$. Doing this for each term, we obtain:

$8x^3-14x^2+6x=2x(4x^2-7x+3 ).$

For many types of problems, it is easier to work with this factored form.

As another example, consider $6x+3$. The common factor is $3$. When factoring $3$ from $6x$, $2x$ is left. When factoring $3$ out of $3$, $1$ remains:

$6x+3=3(2x+1)$.

There are two key points to understand about this kind of factoring:

1. This is the simplest kind of factoring. Whenever trying to factor a complicated expression, always begin by looking for common factors that can be pulled out.
2. The factor must be common to all the terms. For instance, $8x^3-14x^2+6x+7$8 has no common factor, since the last term, $7$, is not divisible by $2$ or $x$.

Perfect Squares

The second type of factoring is based on the "squaring" formulae:

$(x+a)^2=x^2+2ax+a^2$

$(x-a)^2=x^2-2ax+a^2$

For instance, if the problem is $x^2 + 6x + 9$, then one may recognize the signature of the first formula: the middle term is three doubled, and the last term is three squared. Thus, this simplifies to $(x+3)^2$. Other examples are

$x^2+10x+25=(x+5)^2$

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

If the middle term is negative, then the second formula is:

$x^2-8x+16=(x-4)^2$

$x^2-14x=49=(x-7)^2$

This type of factoring only works in this specific case: the middle number is something doubled, and the last number is that same value squared. Furthermore, although the middle term can be either positive or negative, the last term cannot be negative. This is because if a negative is squared, the answer is positive. 

To use this method of factoring, one must keep their eyes open to recognize the pattern. The best way to do this is practice.

Difference Between Two Squares

The third type of factoring is based on the third of the basic formulae:

$(x+a)(x-a)=x^2-a^2$

This formula can be run in reverse whenever subtracting two perfect squares. For instance, if there is $x^2-25$, it can be seen that both $x^2$ and $25$ are perfect squares. Therefore it factors as $(x+5)(x-5)$. Other examples include:

$x^2-64=(x+8)(x-8)$

$16y^2-49=(4y+7)(4y-7)$

$2x^2-18=2(x^2-9)=2(x+3)(x-3)$

Note that, in the last example, the first step is done by pulling out a factor $2$, and there are two perfect squares left. This follows the rule: always begin by pulling out common factors before trying anything else.

Note also that when we are working with real numbers, all positive numbers are squares. So 

$x^2 - 3 = (x+\sqrt{3})(x-\sqrt{3})$.

It often happens that we can use this method twice (or more):

$\begin{aligned} x^4-81 & = (x^2 + 9)(x^2-9) \\ & = (x^2 + 9)(x+3)(x-3). \end{aligned}$

It is important to note that the sum of two squares cannot be factored. 

As in the case of factoring a perfect square, to use this method one has to keep their eyes open to notice the pattern.

Brute Force Factoring

This is the hardest way to factor a polynomial, but the one we need to use when the other ones do not suffice. In this case, the multiplication that is being reversed is the FOIL method, a mnemonic for multiplying two binomials, reminding: First, Outside, Inside, Last, as shown here: 

FOIL method diagram

Start by multiplying the First terms, then the Outside terms, then the Inside terms, and finally the Last terms. Often, the outside and inside terms can eventually be added together. It is important to understand this method, in order to be able to perform it in reverse.

In general, we can multiply any number of polynomials with any number of terms using the distributive property, of which the foil method is just a special case. 

To see how to use this for factoring, we again try to notice a pattern. For example, since $3+7 = 10$ and $3 \cdot 7 = 21$, we have:

$x^2+3x+7x+21=(x+3)(x+7)$

In general:

$x^2 + (a+b)x + ab = (x+a) (x+b)$

Or, of course:

$z^4 + (c-d)z^2 -cd = (z^2+c)(z^2-d)$

which we can factor again by the previous method if $-c$ or $d$ are positive.

Especially when we think $a$ and $b$ are integers, the best tactic to do this is checking for positive and negative factors of the last term, since there are only a limited number of them.