Concept Check When directed to completely factor the polynomial $4x^2{y}^5-8x{y}^3$,a student wrote $2x{y}^3(2x{y}^2-4)$. When the teacher did not give him full credit, he complained because when his answer is multiplied out, the result is the original polynomial. Give the correct answer.

Factoring polynomials involves expressing a polynomial as a product of its factors. The goal is to break down the expression into simpler polynomials or monomials that multiply to give the original. Complete factoring means factoring out the greatest common factor (GCF) and then factoring any remaining expressions further if possible.

The GCF of terms in a polynomial is the largest expression that divides each term without a remainder. Identifying the GCF is the first step in factoring, as it simplifies the polynomial and reveals further factoring opportunities. For example, in 4x^2y^5 - 8xy^3, the GCF is 4xy^3.

After factoring, multiplying the factors back should yield the original polynomial. However, correct factoring requires the factors to be fully simplified and factored completely. Partial factoring, even if it multiplies back correctly, may miss further factorization steps, which is why full credit depends on complete factorization.