1Cademy - Factoring $$5x^3

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Factoring the Greatest Common Factor from a Polynomial

Example

Factoring $5x^3 - 25x^2$

Factor $5x^3 - 25x^2$ by extracting the greatest common factor, which in this case includes a variable component.

Step 1 — Find the GCF of $5x^3$ and $25x^2$ : Factor each term into primes and expanded variables: $5x^3 = 5 \cdot x \cdot x \cdot x$ and $25x^2 = 5 \cdot 5 \cdot x \cdot x$ . The factors common to both are one $5$ and two $x$ s, so $\text{GCF} = 5 \cdot x \cdot x = 5x^2$ .

Step 2 — Rewrite each term as a product of the GCF: Express $5x^3 = 5x^2 \cdot x$ and $25x^2 = 5x^2 \cdot 5$ , giving $5x^2 \cdot x - 5x^2 \cdot 5$ .

Step 3 — Factor out the GCF: $5x^2(x - 5)$ .

Step 4 — Check by multiplying: $5x^2(x - 5) = 5x^2 \cdot x - 5x^2 \cdot 5 = 5x^3 - 25x^2$ ✓.

The factored form is $5x^2(x - 5)$ . This example shows that the GCF of a polynomial can be a monomial containing both a numerical factor and a variable factor. When finding the GCF of expressions with variables, include each shared variable raised to the lowest power present across all terms — here, $x^2$ (the lower of $x^3$ and $x^2$ ).

Updated 2026-04-21

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OpenStax Elementary Algebra

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