Factoring is the process of breaking down a unified whole into the product of its pieces. For example:
can be factored into
This particular case is also the prime factorization of the number. (Prime factorization is useful for finding both the greatest common factor and the least common multiple of a set of numbers).
When factoring an expression, always begin by looking for a greatest common factor among the term(s). The greatest common factor (GCF) can be “pulled out” of every term to create the product of the GCF with the remaining terms:
The GCF is , and when pulled out (divided) from each term, the resulting product is: . Notice that when the is distributed through the parenthesis, the original expression is attained.
Polynomials can sometimes be factored, and there are a few common forms that should be memorized:
, known as a difference of squares
, known as a sum of cubes
, known as a difference of cubes
Again notice that the distribution of the products on the right hand side yields the polynomials on the left hand side.
When provided a trinomial of the form: , where , the factors will be composed of binomials that contain 2 integers that multiply to and combine to . The signs of the coefficients and can help in quickly narrowing the possibilities of the integer values.
If is negative, only one of the integers can be negative. Otherwise, both of the integers are negative or both of the integers are positive.
If is negative, the absolute value of the negative integer is larger than the positive integer. If is positive, the positive integer is larger than the absolute value of the negative integer.
For example: Factor
Because both and are negative, there will be one positive integer and one negative integer. First look for the factors of : . Only and can combine to produce a , so the factored form of the expression is: . Apply the distributive property to verify the factors.
In cases where the coefficient is not equal to or , use this method of factoring but consider the factors of the coefficient as well. The product of the first terms of the binomials must equal .
For example: Factor
Notice again that the and coefficients are positive, so both of the integer factors will be positive. The factors of : and the factors of : .
Try but notice that does not equal so this cannot be the factored form.
Try , which works out to be the answer.
In cases involving a 3rd degree polynomial that contains 4 terms, factoring by grouping might be necessary.
This expression can be rewritten as , and the binomials inside the parentheses can be factored into . Notice that the expression is common to both terms. Consequently, it can be factored out of both terms to yield .