50.5 Perfect Square Trinomial Factoring

Perfect Square Trinomial Factoring is the procedure applied once a trinomial has been confirmed to match the perfect-square pattern, extracting the square-root bases of its outer terms and assembling them into a single squared binomial whose sign matches the sign of the original middle term. It converts the recognition step's confirmation into an actual factored result, handling both plain and composite bases, including those with numerical coefficients and variable exponents, so the technique applies to trinomials of varying complexity.

Because the recognition step has already confirmed the required structural relationship among the three terms, this factoring procedure proceeds directly to assembly without any additional searching.

Choosing the Binomial Sign

Positive-Middle Binomial Square

When the confirmed trinomial's middle term is positive, the binomial factor is assembled using addition between its two extracted bases, matching the sum-square pattern.

Negative-Middle Binomial Square

When the confirmed trinomial's middle term is negative, the binomial factor is assembled using subtraction between its two extracted bases instead, matching the difference-square pattern.

Handling Composite Bases

Composite Monomial Square Bases

When the outer terms of the trinomial are composite expressions containing both a coefficient and a variable factor, each base is extracted by processing the numerical and variable portions separately, exactly as in the numerical and variable component handling used for special products in the forward direction.

Numerical Coefficient Square Roots

The coefficient of a composite outer term is reduced to its square root, becoming the numerical portion of that term's extracted base.

Variable Exponent Halving for Square Bases

Any variable factor of a composite outer term has its exponent divided by two, following the inverse of the power-of-a-power rule, becoming the variable portion of that term's extracted base.

Assembling and Finalizing

Binomial Square Factor Assembly

The numerical and variable portions of each extracted base are recombined into the full base expression, and the two full bases, first and second, are joined with the sign matching the middle term's sign to form the complete binomial being squared.

Repeated Binomial Factor Representation

Since squaring a binomial means multiplying it by itself, the completed factorization can equivalently be written as the same binomial factor listed twice in a product, rather than using exponent notation, though the squared notation is generally preferred for compactness.

Perfect-Square Factorization Completion

The completed factorization is verified by expanding the squared binomial back out using the binomial square identity and confirming the result matches the original trinomial exactly, term for term.