49.9 Trinomial Factorization Verification

Trinomial Factorization Verification ensures accurate factoring of quadratic expressions by testing if the product of factors matches the original trinomial.

Trinomial Factorization Verification is the set of checks used to confirm that a claimed pair of binomial factors, once multiplied back together and combined with any exterior common factor, exactly reproduces the original trinomial that was factored. Because trinomial factoring involves multiple stages, an initial GCF extraction, a search for an integer pair, and the assembly of two binomial factors, verification provides a way to confirm the entire chain of steps by checking the final product against the starting expression.

This verification relies on the distributive property in the same way any factoring check does, expanding the claimed factors fully and comparing the result term by term against the original trinomial.

Expanding the Claimed Factors

Binomial Factor Product Expansion

The two binomial factors are multiplied together using the full distributive property, producing four partial products that will be combined and compared against the original trinomial's three terms.

(x + 2) (x + 3) = x^{2} + 3 x + 2 x + 6

Checking Each Term of the Result

Quadratic Leading-Term Recovery

The product of the two binomials' leading terms is checked against the squared term of the original trinomial, confirming that this first partial product alone accounts for the trinomial's leading term.

x \cdot x = x^{2}

Linear Middle-Term Recovery

The two cross terms produced by the expansion are combined and checked against the linear term of the original trinomial, confirming that their sum matches the required middle coefficient.

3 x + 2 x = 5 x

Constant-Term Product Recovery

The product of the two binomials' constant terms is checked against the constant term of the original trinomial, confirming that this final partial product matches exactly.

2 \cdot 3 = 6

Accounting for Any Exterior Factor

Exterior GCF Recovery

If an overall greatest common factor was extracted before the trinomial was factored, that factor is multiplied back across the fully expanded binomial product as a final step, since the complete factorization must account for this exterior factor in addition to the two binomials.

2 (x + 2) (x + 3) = 2 (x^{2} + 5 x + 6) = 2 x^{2} + 10 x + 12

Confirming the Complete Match

Original Trinomial Agreement

The fully expanded expression, including any exterior factor, is compared term by term against the original trinomial exactly as it was given at the start of the factoring process; a match across every term confirms the factorization is correct.

Complete Integer Factorization Check

Beyond confirming correctness, a final check confirms that the factorization is complete: every coefficient appearing in the two binomial factors and any exterior factor is an integer, and no further common factor remains inside either binomial, matching the integer-coefficient goal that defines the scope of trinomial factoring.

Worked Verification Example

Applying the Full Check

Given the claimed factorization 2x² + 10x + 12 = 2(x + 2)(x + 3), expanding (x + 2)(x + 3) gives x² + 5x + 6, and multiplying by the exterior factor 2 gives 2x² + 10x + 12, which matches the original trinomial exactly, confirming the factorization.

2 (x + 2) (x + 3) = 2 x^{2} + 10 x + 12