Problem 101 - College Algebra

Question

Using an example, explain how to factor out the greatest common factor of a polynomial.

Step-by-Step Solution

Answer

The polynomial \(4x^3 + 8x^2\) factors to \(4x^2(x + 2)\) using the GCF of \(4x^2\).

Step 1: Identifying the Polynomial and its Terms

Let's take the polynomial \(4x^3 + 8x^2\). This polynomial has two terms - \(4x^3\) and \(8x^2\). Each term is a product of a numerical coefficient and a power of \(x\).

Step 2: Finding the Greatest Common Factor

The next step is to identify the greatest common factor (GCF) of the terms of the polynomial. In this case, the GCF is \(4x^2\), which is the highest factor that is common to both terms.

Step 3: Factoring Out the GCF

Next, each term of the polynomial is divided by the GCF, resulting in a factored polynomial. Thus, \(4x^3\) divided by \(4x^2\) gives \(x\), and \(8x^2\) divided by \(4x^2\) gives \(2\). The fully factored polynomial is therefore \(4x^2(x + 2)\).