Factor completely, or state that the polynomial is prime. $5x^3-45x$

Verified step by step guidance

1. Identify the greatest common factor (GCF) of the terms in the polynomial \$5x^3 - 45x$. The GCF is the largest expression that divides both terms evenly.
2. Factor out the GCF from each term. Since the GCF is \$5x\(, rewrite the polynomial as \)5x(\ldots)$.
3. Divide each term by the GCF to find the remaining factor inside the parentheses: \$5x^3 \div 5x = x^2\( and \)-45x \div 5x = -9$.
4. Write the factored form as \$5x(x^2 - 9)$.
5. Recognize that \(x^2 - 9\) is a difference of squares, which can be factored further using the formula \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 3\), so factor it as \((x - 3)(x + 3)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials or factors. This process helps simplify expressions and solve polynomial equations. Common methods include factoring out the greatest common factor, grouping, and special products.

Greatest Common Factor (GCF)

The greatest common factor is the largest expression that divides all terms of a polynomial without leaving a remainder. Factoring out the GCF is often the first step in factoring polynomials, simplifying the expression and making further factoring easier.

Prime Polynomials

A prime polynomial is one that cannot be factored further over the set of real numbers. After attempting all factoring methods, if no factors other than 1 and the polynomial itself are found, the polynomial is considered prime.