Factoring Out the Greatest Common Factor (GCF) from a Polynomial

Q1. Factor out the GCF from the polynomial:

$\frac{1}{5}x^4 + \frac{4}{5}x^6 - \frac{6}{5}x^5 + \frac{1}{5}x$

Background

Topic: Factoring Polynomials

This question is testing your ability to identify and factor out the greatest common factor (GCF) from all terms in a polynomial. Factoring is a fundamental skill in algebra that simplifies expressions and prepares them for solving equations.

Key Terms and Formulas

• Greatest Common Factor (GCF): The largest expression that divides each term of the polynomial.

• Factoring: Writing a polynomial as a product of its GCF and another polynomial.

To factor out the GCF, look for common numerical coefficients and variable factors in each term.

Step-by-Step Guidance

1. Examine each term to identify common numerical and variable factors. All terms have a coefficient with $\frac{1}{5}$ and a variable $x$ raised to a power.

2. Determine the GCF for the coefficients. Since all coefficients are multiples of $\frac{1}{5}$, this is the numerical GCF.

3. Determine the GCF for the variable part. Each term contains $x$ raised to a power; the lowest power is $x$.

4. Write the GCF as $\frac{1}{5}x$ and factor it out from each term. Set up the expression as $\frac{1}{5}x(\ldots)$, where the parentheses will contain the remaining terms after factoring.

Try solving on your own before revealing the answer!