In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x^3−8a^2 x+24x^2+72x

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common methods include factoring out the greatest common factor (GCF), using the difference of squares, and applying the quadratic formula for trinomials. Understanding these techniques allows for the complete factorization of polynomials.

Greatest Common Factor (GCF)

The greatest common factor (GCF) is the largest factor that divides all terms in a polynomial. Identifying the GCF is often the first step in factoring, as it simplifies the polynomial and makes it easier to work with. For example, in the polynomial 2x^3−8a^2 x+24x^2+72x, the GCF can be factored out to simplify the expression before further factoring.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two non-constant polynomials with real coefficients. Recognizing prime polynomials is crucial in algebra, as it indicates that the polynomial is in its simplest form. In the context of the given polynomial, determining whether it is prime or can be factored completely is essential for solving the exercise.