Factoring

Factoring is the process of putting a polynomial into a form where the solutions and roots of the polynomial are clear. There is no systematic way of factoring polynomials other than estimated guess and check. This article will discuss various strategies for factoring polynomials as well as some special situations.

# Factored Form

An expression is in factored form when all of the factors have been exposed. So, an expression in factored form is simply a bunch of parts that multiply together in order to get the original expression. The factored form of 7x2+9x+2 is (7x+2)(x+1) because the factored form when simplified yields the original expression.

To look at this with a simpler example, factor 124. When you factor a constant, you attempt to extract all of the prime factors. So, we need to find the prime factorization of 124. Using a quick factor tree, we get (2)(2)(31), or (2)2(31). Those factors do not have any more factors that can be extracted, so the number 124 is fully factored.

Essentially, this is what you are doing when you are factoring an expression. You are extracting prime factors. So, when you factor something like x2-49, you try to find the factors that can't be factored. The factored form of that expression is (x+7)(x-7) because the two factors can't be factored themselves and when multiplied result with x2-49.

# The Process

To factor a polynomial, there is a process that you can undergo. There is no actual systematic way to factor polynomials, and there is certainly no guaranteed quick solution. When factoring, you want to search for a GCF first. Next, attempt to search for special circumstances, and finally commence guess and check.

# GCF

GCF stands for the greatest common factor. It is the largest factor of a group of numbers common to each number in the group. So, let's say we has a 10 and a 25. You can find the greatest common factor by listing the factors in each problem. The prime factors of 10 are 2*5. The prime factors of 25 are 5*5. Now, take the numbers that are common for each. In this case, each has at least one 5, so we can extract 5. That means the GCF of 10 and 25 is 5.

The GCF of an expression is very much the same way. For example, take 4x2 and 6x. Start by listing their prime factors remembering that 4x2 is really 4*x*x. For 4x2, we get 2*2*x*x. For 6x, we get 2*3*x. Each has one 2 and one x in common. Take those out and multiply. The GCF of 4x2 and 6x is 2x.

The GCF helps when factoring expressions. Factor 4x2+6x. When factoring, the first step is always to extract all GCF's. So, we already know the GCF is 2x. Remembering the distributive property, we can actually divide each term by the 2x and then multiply that 2x by the sum of the quantity. That means that 4x2+6x factors to 2x(2x+3).

# Special Circumstances

There are a few special occurrences that turn out to be easily factorized. These occurrences have very specific patterns that can be identified.

A perfect quadratic is one such expression in the terms a2+2ab+b2 or a2-2ab+b2.