In Exercises 23–48, factor completely, or state that the polynomial is prime. 2x³ - 8x

Verified step by step guidance

1. Step 1: Identify the greatest common factor (GCF) of the terms in the polynomial. In this case, the GCF of \(2x^3\) and \(-8x\) is \(2x\).
2. Step 2: Factor out the GCF from each term in the polynomial. This means you will divide each term by \(2x\) and write the polynomial as \(2x(x^2 - 4)\).
3. Step 3: Observe the expression inside the parentheses, \(x^2 - 4\). Notice that it is a difference of squares, which can be factored further.
4. Step 4: Apply the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), to \(x^2 - 4\). Here, \(a = x\) and \(b = 2\), so \(x^2 - 4 = (x - 2)(x + 2)\).
5. Step 5: Combine the factored terms to express the completely factored form of the polynomial: \(2x(x - 2)(x + 2)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process often includes identifying common factors, applying special factoring techniques like difference of squares, or using methods such as grouping. Understanding how to factor is essential for simplifying expressions and solving polynomial equations.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that divides two or more numbers or terms without leaving a remainder. In polynomial expressions, finding the GCF allows for simplification by factoring it out, which can make the remaining polynomial easier to work with. Recognizing the GCF is a crucial first step in the factoring process.

Prime Polynomials

A prime polynomial is a polynomial that cannot be factored into simpler polynomials with rational coefficients. Identifying whether a polynomial is prime is important in algebra, as it determines whether further simplification is possible. Understanding the criteria for primality helps in recognizing when a polynomial is already in its simplest form.