In Exercises 65–92, factor completely, or state that the polynomial is prime. 2x³−8a²x+24x²+72x

Verified step by step guidance

1. Group the terms of the polynomial into two pairs to make factoring easier: \( (2x^3 - 8a^2x) + (24x^2 + 72x) \).
2. Factor out the greatest common factor (GCF) from each group: \( 2x(x^2 - 4a^2) + 24x(x + 3) \).
3. Notice that \( x^2 - 4a^2 \) is a difference of squares. Factor it as \( (x - 2a)(x + 2a) \), so the expression becomes \( 2x(x - 2a)(x + 2a) + 24x(x + 3) \).
4. Factor out the common term \( x \) from the entire expression: \( x[2(x - 2a)(x + 2a) + 24(x + 3)] \).
5. Simplify the expression inside the brackets and check if further factoring is possible. Combine like terms and verify if the polynomial is fully factored or if it is prime.

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 techniques include factoring out the greatest common factor (GCF), using special products like the difference of squares, and applying methods such as grouping or the quadratic formula for higher-degree 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 cannot be simplified further. In the context of the given polynomial, determining whether it is prime or can be factored completely is essential for solving the exercise.