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What is the completely factored form of $x y^3-x^3 y$?


A$x y(x-y)\left(x^2+x y+y^2\right)$
B$x y(y+x)(y-x)$
C$x y(y-x)(y-x)$
D$x y(x-y)\left(y^2+x y+x^2\right)$

Verified Answer :

  • First, factor out the common term \(xy\) from the expression \(xy^3 - x^3y\) to get \(xy(y^2 - x^2)\).
  • Then, apply the difference of squares identity \(y^2 - x^2 = (y - x)(y + x)\) to further decompose the expression.
  • Combine these parts to reach the completely factored form: \(xy(y - x)(y + x)\).
  • Conclude that the correct choice is option b. The final answer is \(\boxed{b}\).

Explanation

  1. Factoring out the Greatest Common Factor

The expression to factor is \(xy^3 - x^3y\). We begin by identifying the greatest common factor (GCF) of the two terms. Both terms contain at least one \(x\) and one \(y\), so we can factor out \(xy\). Dividing each term by \(xy\) gives:
\[xy^3 - x^3y = xy(y^2 - x^2)\]

  1. Applying the Difference of Squares Identity

Next, we analyze the expression inside the parentheses: \(y^2 - x^2\). This is a classic difference of squares, which follows the algebraic identity \(A^2 - B^2 = (A - B)(A + B)\). Applying this identity with \(A = y\) and \(B = x\), we get:
\[y^2 - x^2 = (y - x)(y + x)\]

  1. Final Conclusion and Comparison with Options

Substituting the factored form of the binomial back into our expression, we obtain:
\[xy(y^2 - x^2) = xy(y - x)(y + x)\]
Comparing this result to the given options:

A\(xy(x - y)(x^2 + xy + y^2)\)
B\(xy(y + x)(y - x)\)
C\(xy(y - x)(y - x)\)
D\(xy(x - y)(y^2 + xy + x^2)\)

Option (b) is equivalent to our result, as multiplication is commutative: \(xy(y + x)(y - x)\) is the same as \(xy(y - x)(y + x)\).

Examples

Factoring polynomials is a fundamental skill in algebra used to simplify equations, solve for variables, and identify function roots. For example, if you are designing a product's packaging and the volume is modeled by a polynomial like \(V = xy^3 - x^3y\), factoring helps you determine the dimensions of the base and height more easily by breaking the expression into simpler linear factors. This is essential for optimizing material usage and structural efficiency in manufacturing and engineering.

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