Factoring Polynomials: A Step-by-Step Guide

Factoring polynomials is a fundamental skill in algebra and is essential for solving complex equations. This guide will walk you through the process of factoring a specific polynomial, demonstrating various techniques that can be applied more broadly. We'll focus on the polynomial x^2 5x - 2x - 10 and explore multiple methods to simplify it.

Understanding the Polynomial

The polynomial we are working with is:

x^2 5x - 2x - 10

This polynomial is a quadratic expression, which means it is a second-degree polynomial. The standard form of a quadratic equation is:

ax^2 bx c

In our case, a 1, b 3, and c -10. Let's apply the methods to factor this polynomial.

Factoring by Grouping

Method 1: Grouping

Factoring by grouping involves splitting the polynomial into two or more groups, making it easier to find common factors in each group. Here's how we can do it:

Start by identifying the polynomial: x^2 5x - 2x - 10 Rearrange the terms to group them: x^2 5x - 2x - 10 x^2 3x - 2x - 10 Group the first two terms and the last two terms: (x^2 3x) (-2x - 10) Factor out the greatest common factor (GCF) from each group: From x^2 3x, factor out x: x(x 3) From -2x - 10, factor out -2: -2(x 5) Notice that (x 3) and (x 5) are not the same, so we need to re-evaluate. Let's try another approach.

Method 2: Factoring by Grouping (Corrected)

Let's correct and re-arrange:

Rearrange the terms to facilitate grouping: x^2 5x - 2x - 10 x^2 5x - 2x - 10 Group the first two terms and the last two terms: (x^2 5x) (-2x - 10) Factor out the GCF from each group: From x^2 5x, factor out x: x(x 5) From -2x - 10, factor out -2: -2(x 5) Notice that the binomials are the same: (x 5). So, we can pull this common factor out: The factored form is: (x 5)(x - 2)

Conclusion

By using the grouping method, we successfully factored the polynomial x^2 5x - 2x - 10 into (x 5)(x - 2).

Further Application

This method can be applied to other polynomials as well. Here are some further examples and how to solve them:

Example 1: Factoring x^2 4x 4

Rearrange and group terms if needed. Factor out the GCF from each group if applicable. Identify the common factor if any. The factored form is: (x 2)^2.

Example 2: Factoring x^3 - 3x^2 - 4x 12

Group terms and factor out the common factor in each group. Identify the common binomial factor. The factored form is: (x - 3)(x 2)(x - 2).

Conclusion

Factoring polynomials is a valuable skill in algebra. By using the grouping method, you can simplify and solve complex polynomial equations. Practice with different polynomials to master this technique and apply it to other areas of mathematics.