48.5 Factoring Out the GCF

Factoring Out the GCF is the complete procedure that takes a constructed monomial greatest common factor and applies it to an entire polynomial, producing a factored form consisting of the GCF multiplied by a simplified remaining polynomial. It picks up where monomial GCF construction leaves off, using the already identified common factor to divide through the original expression term by term and assemble the two pieces, exterior factor and interior polynomial, into the final factored product.

This procedure is the practical endpoint of common-factor factoring: it transforms a sum of terms that share a common factor into an equivalent product, which is often a more useful form for solving equations or simplifying expressions further.

Starting from the Identified Factor

Polynomial GCF Extraction

The process begins with the monomial GCF already determined for the polynomial in question, treating it as the fixed value by which every term of the polynomial will now be divided, one term at a time.

Dividing Each Term

First Term Division by the GCF

The first term of the polynomial is divided by the GCF, applying ordinary coefficient division and the exponent subtraction rule to its variable factors, producing the first term of what will become the interior polynomial.

Remaining Term Division by the GCF

Every subsequent term of the polynomial is divided by the same GCF in turn, using the identical process, until every original term has been converted into its corresponding quotient term.

Quotient Polynomial Formation

The individual quotient terms produced from dividing each original term by the GCF are combined, preserving their original signs, to form the interior polynomial that will sit inside the parentheses of the factored form.

Assembling the Factored Form

Exterior GCF and Interior Polynomial Product

The GCF is written outside a set of parentheses, and the interior polynomial formed from the quotient terms is written inside, expressing the original polynomial as this product.

Simplifying and Checking the Interior

Interior Polynomial Term Simplification

Each term inside the parentheses is checked to confirm it has been fully reduced, meaning its coefficient and any variable exponents are as simple as the division allows, with no further simplification remaining for any single term.

Residual Common Factor Inspection

The interior polynomial is inspected once more to confirm that no additional common factor remains among its own terms; if a further shared factor is found, it indicates the original GCF extracted was not actually the greatest common factor, and the process should be revisited using the correctly maximized GCF.

The Final Result

GCF-Factored Polynomial Form

The completed factored form presents the original polynomial as a single product of its greatest common factor and a fully reduced interior polynomial, and this form can always be checked by redistributing the exterior factor back across the interior terms to confirm it reproduces the original expression exactly.