how to factor x 3 x 2

2 min read 10-01-2025

Factoring cubic polynomials like x³ + x² can seem daunting, but it's manageable with the right approach. This guide will walk you through the steps, explaining the concepts clearly and providing examples. We'll focus on finding the greatest common factor (GCF) and then exploring any further factoring possibilities.

Understanding Factoring

Factoring a polynomial means rewriting it as a product of simpler expressions. Just like factoring a number (e.g., 12 = 2 x 2 x 3), we break down a polynomial into its multiplicative components. This is a crucial skill in algebra, used extensively in solving equations and simplifying expressions.

Step-by-Step Factoring of x³ + x²

1. Identify the Greatest Common Factor (GCF):

The first step in factoring any polynomial is to look for a greatest common factor (GCF) among the terms. In x³ + x², both terms contain an x. The lowest power of x present is x², so x² is our GCF.

2. Factor out the GCF:

Now, we factor out the GCF (x²) from both terms:

x³ + x² = x²(x + 1)

This is the factored form of x³ + x². We've successfully broken down the original expression into a product of two simpler expressions: x² and (x + 1).

Checking Your Work

It's always a good idea to check your answer. You can do this by expanding your factored form:

x²(x + 1) = x² * x + x² * 1 = x³ + x²

This matches our original expression, confirming that our factoring is correct.

Factoring Polynomials with More Terms or Complex Expressions

While x³ + x² factored easily using the GCF, other cubic polynomials might require more advanced techniques. These include:

Grouping: Useful for polynomials with four or more terms. You group terms with common factors and then factor out the common factor from each group.
Sum or Difference of Cubes: Specific formulas exist for factoring expressions in the form a³ + b³ or a³ - b³.
Factoring by Substitution: This involves replacing parts of the expression with a simpler variable to make factoring easier.

Example: Factoring 2x³ + 4x²

Let's apply the same process to a slightly more complex example: 2x³ + 4x².

Find the GCF: The GCF of 2x³ and 4x² is 2x².
Factor out the GCF:

2x³ + 4x² = 2x²(x + 2)
Check: Expanding 2x²(x + 2) gives us 2x³ + 4x², confirming the factoring.

Conclusion

Factoring x³ + x², and similar cubic polynomials, primarily involves identifying and factoring out the greatest common factor. This simplifies the expression and allows for further manipulation if needed. Remember to always check your answer by expanding the factored form to ensure accuracy. Mastering these basic factoring techniques is foundational to success in algebra and beyond.