Factoring Polynomials - Common Monomial Factoring Grade 8

Introduction

This tutorial covers the process of factoring polynomials using common monomial factoring. Designed for Grade 8 students, this guide breaks down the steps to help you understand how to identify and factor out the greatest common factor (GCF) from polynomial expressions. Mastering this skill is essential for solving more complex algebraic problems.

Step 1: Identify the Greatest Common Factor

To start factoring a polynomial, you need to find the GCF of the terms in the expression.

• List the coefficients: Write down the numerical coefficients of each term.
• Find the GCF: Determine the GCF of these coefficients. For example, for the coefficients 6, 12, and 18, the GCF is 6.
• Identify the variable factors: Look at the variables in each term. For instance, if the terms are (x^3), (x^2), and (x), the GCF of the variable part is (x).
• Combine the GCF: Multiply the numerical GCF by the variable GCF to find the overall GCF for the polynomial.

Step 2: Factor Out the GCF

Once the GCF is identified, the next step is to factor it out from each term of the polynomial.

• Rewrite the polynomial: Express the polynomial as the product of the GCF and another polynomial.
• Distribute the GCF: For example, if you have (6x^3 + 12x^2 + 18x), factor it as follows:
  • GCF is (6x).
  • Rewrite as (6x(x^2 + 2x + 3)).

Step 3: Simplify the Remaining Polynomial

After factoring out the GCF, focus on simplifying the remaining polynomial if possible.

• Check for further factoring: Examine the polynomial inside the parentheses to see if it can be factored further.
• Use factoring techniques: Techniques such as grouping, difference of squares, or quadratic factoring may apply based on the polynomial structure.

Step 4: Verify Your Work

To ensure that your factoring is correct, check your results.

• Expand the factors: Multiply the GCF back with the polynomial you factored out.
• Compare with original: Ensure that the expanded expression matches the original polynomial.

Conclusion

Factoring polynomials using common monomial factoring is a crucial skill in algebra. By identifying the GCF, factoring it out, simplifying the remaining polynomial, and verifying your work, you can solve polynomial expressions more effectively. Practice with various polynomial examples to enhance your understanding and proficiency in factoring. For your next steps, explore factoring polynomials that do not have a common monomial or delve into quadratic equations for deeper comprehension.