48.8 Factoring Verification

Factoring Verification is the collection of checks used to confirm that a factored expression is genuinely equivalent to the original polynomial it was derived from, and that the factorization has been carried out as completely as possible. Because factoring is the reverse of multiplication, its most direct verification method is to multiply the factored form back out and compare the result against the original polynomial, though several structural checks can also catch specific classes of error without requiring a full redistribution.

These checks apply across the different factoring techniques, common-factor extraction and factoring by grouping alike, since both ultimately produce a product of factors that must reconstruct the same original expression.

Reversing the Factorization

Factored Expression Distribution

The most direct verification method distributes the factored expression back out, multiplying the exterior factor across every term of the interior polynomial, exactly reversing the extraction process that produced the factored form.

Original Polynomial Reconstruction

The result of this distribution is compared term by term against the original polynomial that was factored; if the two expressions match exactly, the factorization is confirmed correct.

Term-Level Checks

Every Original Term Recovery

Each term of the original polynomial should reappear somewhere in the redistributed expression; a missing term after redistribution indicates that a term was dropped somewhere during the factoring process, most likely during the initial GCF extraction.

Coefficient and Sign Agreement

Each redistributed term's coefficient, including its sign, is checked against the corresponding original term's coefficient; a coefficient that comes out with the wrong magnitude or the wrong sign points to an arithmetic slip during either the extraction or the redistribution.

Polynomial Degree Agreement

The degree of the redistributed expression is checked against the degree of the original polynomial; since factoring does not change the overall degree of an expression, any mismatch here signals an error in how variable exponents were handled during extraction.

Checking Completeness

Residual GCF Absence Check

Beyond confirming correctness, this check confirms completeness: the interior polynomial left after extraction is inspected to ensure no further common factor remains among its own terms, since a leftover common factor would mean the original extraction did not use the true greatest common factor.

Verifying a Grouping Factorization

Grouping Product Reconstruction

For a polynomial factored by grouping, verification multiplies the two resulting binomial factors together using the distributive property in full, producing four partial products that should recombine into the original four-term polynomial.

Factorization Value Agreement

As a further independent check, a numerical value can be substituted into both the original polynomial and the factored expression; because factoring preserves equivalence for every input value, both expressions must evaluate to the same number, and any disagreement reveals an error somewhere in the factorization.

Worked Verification Example

Applying the Full Set of Checks

Given the claimed factorization 6x² + 15x = 3x(2x + 5), redistribution gives 6x² + 15x, which matches the original exactly; the interior binomial 2x + 5 has no further common factor between its two terms, confirming the factorization is both correct and complete.