What is factoring? Let's ask professionals
So what is factoring in algebra? Factoring or factorization of a polynomial equation is the process of deriving that equation as products of other polynomials. The immediate question that would arise in mind would be what are factors? Each small products obtained can be evaluated for finding the value of variable and this values are known as factors of given expression. The same rule applies for numbers as well as matrices. The main purpose behind doing so is to reduce them to basic forms. The process reverse to that of factorization is expansion where we multiply all the product values to yield the original expression.
Ways of achieving factorization:
Now that we have learned what is factoring in maths, we will go for the techniques of factoring algebra. Normally factorization can be achieved in three different ways as follow:

finding difference of two square variables.

grouping and taking the common part out.

dividing the expression in valid perfect trinomials.
If we have a equation of the form x^{2} – y^{2}, then it can be rewritten as the product (xy)(x+y). if we multiply those products, we get the original equation back. Taking the factors out by grouping method requires to select the the common part and group them accordingly. It could be a bit tricky as well. Consider the expression x^{2} + 2x 3. Here we need to group it in a manner that both the groups contain common part. Lets write it as x^{2} + 3x – x – 3 = x(x + 3) – 1(x + 3) = (x + 3)(x  1). Thus right hand side of the equation can be solved to find values of x. Another method is about perfect square or cube expressions. If we have expression of the form x^{2} – 2xy + y^{2}, it can be directly written as (x – y)^{2}. Similarly for cubical expressions, (x + y)^{3} can be written instead of x^{3} + y^{3} + 3x^{2}y + 3xy^{2}. This forms can be mugged up and used wherever required. Some examples are required to get clear through the topic.
(1) Find the factors for expression a^{2} – 9:
This one can be expressed ad different of squares of a and 3 i.e. (a – 3)(a + 3). Hence solving for a, we have a = 3 or 3.
(2) Evaluate a^{2} + 2a + 1=0:
This can be expressed as sum of squares (a + 1)^{2}. Hence a = 1.
(3) Find a if a^{2} – 3a – 4=0:
Lets group it in a proper way to take the common part out. a^{2} – 3a – 4=0 => a^{2} – 4a + a – 4 = 0. Hence a(a – 4) + 1(a – 4) = 0 or (a + 1)(a – 4) = 0 which yields a = 1 or a = 4.
Thus we have learned the process of factorization of algebraic equations and finding the factors along with examples. Students should go through practice problems and make themselves master of factors.
More topics in Factoring  
Greatest Common Factor  Greatest Common Factor Examples 
Find The Greatest Common Factor 
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