For all real numbers x and y such that x \neq -y and x \neq -2y, which of the following ex… — GRE General Test

For all real numbers

x

and

y

such that

x \neq -y

and

x \neq -2y

, which of the following expressions is equivalent to

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

x - y

Factoring the numerator by grouping gives

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

. Factoring the denominator yields

(x + y)(x + 2y)

. Dividing the numerator by the denominator cancels the common factors

(x + y)

and

(x + 2y)

, leaving

x - y

Factor the numerator by grouping terms

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

Grouping pairs of terms allows factoring out common binomial factors.

Factor the quadratic denominator

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

Finding two terms whose sum is

3y

and product is

2y^2

factors the quadratic in

x

Simplify the rational expression by canceling non-zero common factors

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

Since

x \neq -y

and

x \neq -2y

, the factors

(x + y)

and

(x + 2y)

are non-zero and can be canceled.

Polynomial factoring by grouping, difference of squares, quadratic trinomial factoring, and simplifying rational algebraic expressions.

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