If terms in an expression share a common factor, it can be factored out of the expression. It may be helpful to think of this process as being the opposite of distribution. Here are some examples with a simple algebraic expressions:
As seen in the second example, you can factor out positive or negative factors. However, be sure to check the signs between terms in parentheses to make sure you have correctly factored a common factor between terms.
Factoring out an Algebraic Factor
We can use this factoring technique to factor out more than just numbers. If algebraic expressions share variable factors, we can factor them out as well. Consider the following examples:
It helps to break down coefficients into prime factors. Doing so in the example above makes it clearer to see how 2x is a common factor. Additionally, if an entire term is the factor being factored out, what remains is 1. This is why the binomial in parentheses above is (3x + 1).
Dividing a Polynomial by a Monomial
Sometimes when dividing a polynomial by a monomial, there are common factors between the monomial term and terms that make up the polynomial. In this case, it is relatively straightforward to divide coefficients and decrease exponents. This is illustrated in the example below:
Of course, such examples that divide nicely between all terms are not always the case when dividing polynomials. When we need to divide a term that doesn't share all common factors, we divide what we can, and express the remainder as a fraction.
To show this, we'll adjust our previous example slightly, such that the first couple of terms divide evenly, but the last term does not. Take note as to how we write the division:
We were able to do factor our 3x from –6x3 and 18x2. However, the only common factor between –3 and 3x is the 3, not the x. So this term becomes a fraction, with –3 in the numerator and 3x in the denominator. Due to the common factor of 3, we were able to simplify to –1 divided by x.