For all real numbers x and y such that x \neq y, x \neq -y, and x^2 + y^2 \neq 0, consider… — GRE General Test

For all real numbers $x$ and $y$ such that $x \neq y$ , $x \neq -y$ , and $x^2 + y^2 \neq 0$ , consider the algebraic expression:

E(x, y) = \frac{x^4 - y^4}{x^3 - x^2y + xy^2 - y^3}

Which of the following expressions are equivalent to $E(x, y)$ for all valid values of $x$ and $y$ ? Select all such expressions.

The expressions equivalent to

E(x, y)

are

\frac{x^2 - y^2}{x - y}

and

\frac{x^3 + y^3}{x^2 - xy + y^2}

Simplifying

E(x, y)

by factoring both numerator and denominator yields

x + y

. The option with

\frac{x^2 - y^2}{x - y}

simplifies directly to

x + y

by canceling

(x - y)

. The option with

\frac{x^3 + y^3}{x^2 - xy + y^2}

uses the sum of cubes identity to factor the numerator into

(x + y)(x^2 - xy + y^2)

, which also cancels down to

x + y

. Both of these options are mathematically identical to

E(x, y)

Factor the numerator of

E(x, y)

using the difference of squares identity twice.

x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) = (x - y)(x + y)(x^2 + y^2)

Decomposing higher-power binomials into linear and quadratic factors enables algebraic simplification.

Factor the denominator of

E(x, y)

by grouping terms.

x^3 - x^2y + xy^2 - y^3 = x^2(x - y) + y^2(x - y) = (x - y)(x^2 + y^2)

Grouping adjacent terms with shared factors allows factoring out

(x - y)

Simplify the full expression

E(x, y)

by canceling common non-zero factors.

E(x, y) = \frac{(x - y)(x + y)(x^2 + y^2)}{(x - y)(x^2 + y^2)} = x + y

Since

x \neq y

and

x^2 + y^2 \neq 0

, the factors

(x - y)

and

(x^2 + y^2)

cancel completely.

Evaluate the given options to determine which simplify to

x + y

\frac{x^2 - y^2}{x - y} = x + y

and

\frac{x^3 + y^3}{x^2 - xy + y^2} = x + y

, whereas

\frac{x^3 - y^3}{x^2 + xy + y^2} = x - y

and

\frac{x^2 - y^2}{x + y} = x - y

Matching each option's fully simplified form to

x + y

identifies all valid equivalent expressions.

Simplifying rational expressions by polynomial factoring (grouping, difference of squares, and sum of cubes).

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