How Do You Factor Each Polynomial Completely?

How Do You Factor Each Polynomial Completely?

How Do You Factor Each Polynomial Completely?

Factoring completely is an algebraic process that incorporates the three common factoring techniques: the greatest common factor, trinomials and the difference between two squares. These techniques by themselves often factor a polynomial into a single pattern, while factoring completely utilizes them together to simplify an expression as much as possible.

  1. Factor the greatest common factor from the expression

    In the example, 12x^4 - 3x^2 - 54, the greatest common factor is 3. Therefore, the 3 is removed from the initial equation, and the expression becomes 3(4x^4 - x^2 - 18). If there is no greatest common factor to be found, this step may be skipped.

  2. Factor a trinomial

    If there are more than two terms in an expressions, it may be factored into a trinomial. In the example, 3(4x^4 - x^2 - 18), the expression would become 3(4x^2 - 9)(x^2 + 2) when factored.

  3. Factor a difference between two squares

    Finally, in the example 3(4x^2 - 9)(x^2 + 2), (4x^2 - 9) may be factored as a difference between two squares. The expression then becomes 3(2x + 3)(2x- 3)(x^2 + 2) and is factored completely. Many equations may not be factored as a difference between two squares, in which case, this step may be skipped.