Concept Check When directed to completely factor the polynomial $4 x^{2} y^{5} - 8 x y^{3}$ ,a student wrote $2 x y^{3} (2 x y^{2} - 4)$ . 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.

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Start by identifying the greatest common factor (GCF) of the terms in the polynomial $4x^2y^5 - 8xy^3$. Look at the coefficients, variables, and their exponents separately.

The coefficients are 4 and 8, so the GCF of the coefficients is 4. For the variables, find the lowest powers of $x$ and $y$ common to both terms. The first term has $x^2$ and $y^5$, the second has $x$ and $y^3$. So the GCF for variables is $x^1 y^3$.

Combine the GCF of coefficients and variables to get the overall GCF: \$4xy^3$. Factor this out of the polynomial:

\[4x^2y^5 - 8xy^3 = 4xy^3(\text{?})\]

Divide each term of the original polynomial by the GCF \$4xy^3$ to find the terms inside the parentheses: $\frac{4x^2y^5}{4xy^3} = x y^2$ and $\frac{-8xy^3}{4xy^3} = -2$. So the completely factored form is:

\[4xy^3(x y^2 - 2)\]

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주요 개념

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Factoring Polynomials

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.

Greatest Common Factor (GCF)

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.

Verification by Multiplication

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.