50.8 Special Factorization Verification

Special Factorization Verification ensures mathematical expressions are correctly factored by applying algebraic rules and checking for accuracy in decomposition.

Special Factorization Verification is the set of checks used to confirm that a factorization produced through a special pattern, difference of squares, perfect square trinomial, or sum or difference of cubes, genuinely reproduces the original polynomial when its factors are expanded back out. Because each special pattern involves its own distinct expansion identity, verification applies the specific identity relevant to whichever pattern was used, then compares the expanded result against the original expression term by term.

This verification serves the same fundamental purpose as verification for any other type of factoring: confirming equivalence between the factored form and the starting expression, while also confirming that the chosen pattern was the correct one to apply.

Expanding by Pattern Type

Difference-of-Squares Product Expansion

For a difference-of-squares factorization, the two conjugate binomial factors are multiplied together, and the resulting cross terms should cancel completely, leaving only the two squared terms of the original expression.

(a + b) (a - b) = a^{2} - b^{2}

Binomial-Square Product Expansion

For a perfect-square-trinomial factorization, the squared binomial factor is expanded using the binomial square identity, producing three terms that should match the original trinomial's leading term, middle term, and constant term exactly.

(a + b)^{2} = a^{2} + 2 a b + b^{2}

Cube-Pattern Product Expansion

For a sum-of-cubes or difference-of-cubes factorization, the binomial and trinomial factors are multiplied together in full, and the resulting six partial products should reduce, through cancellation of the four cross terms, down to exactly the original two-term cube expression.

(p + q) (p^{2} - p q + q^{2}) = p^{3} + q^{3}

Comparing Against the Original

Original Polynomial Term Recovery

Every term of the expanded result is compared against the corresponding term of the original polynomial that was factored, confirming that each one, in both magnitude and sign, matches exactly.

Original Polynomial Sign Agreement

Particular attention is paid to matching signs, since a special-pattern factorization applied with the wrong sign, such as using the sum-of-cubes trinomial pattern when the difference-of-cubes pattern was required, produces an expansion that differs from the original only in certain signs, which can otherwise be easy to overlook.

Polynomial Degree Preservation Check

The degree of the expanded result is checked against the degree of the original polynomial, confirming that the special pattern applied did not inadvertently change the overall degree of the expression, since factoring never alters degree.

Reattaching and Rechecking

Special-Pattern GCF Reconstruction

If an overall greatest common factor was extracted before the special pattern was applied, that factor is multiplied back across the pattern's expanded result as a final step, and the complete reconstructed expression is compared against the original polynomial before any GCF was removed.

3 (x + 4) (x - 4) = 3 (x^{2} - 16) = 3 x^{2} - 48

Complete Factorization Reinspection

Finally, every factor in the completed factorization is reinspected to confirm none of them can be broken down any further by another application of a special pattern, ensuring the verified factorization is not only correct but also complete.

Worked Verification Example

Applying the Full Check

Given the claimed factorization 3x² − 48 = 3(x + 4)(x − 4), expanding (x + 4)(x − 4) gives x² − 16, and multiplying by the exterior factor 3 gives 3x² − 48, matching the original polynomial exactly and confirming both correctness and completeness.