For all real numbers x \neq 2, which of the following expressions is equivalent to \frac{x… — GRE General Test

For all real numbers $x \neq 2$ , which of the following expressions is equivalent to $\frac{x^4 - 16}{x^3 - 2x^2 + 4x - 8}$ ?

x + 2

Factoring the numerator

x^4 - 16

as a difference of squares yields

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

. Factoring the denominator

x^3 - 2x^2 + 4x - 8

by grouping yields

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

. Dividing the numerator by the denominator allows the shared factor

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

to cancel, leaving

x + 2

Factor the numerator

x^4 - 16

x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)

Apply the difference of squares factorization identity twice.

Factor the denominator

x^3 - 2x^2 + 4x - 8

by grouping

x^3 - 2x^2 + 4x - 8 = x^2(x - 2) + 4(x - 2) = (x - 2)(x^2 + 4)

Group the terms in pairs and factor out the common binomial factor

(x - 2)

Simplify the rational expression by canceling common factors

\frac{(x - 2)(x + 2)(x^2 + 4)}{(x - 2)(x^2 + 4)} = x + 2

Cancel the non-zero common factor

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

present in both numerator and denominator.

Simplifying rational expressions using difference of squares and factoring by grouping

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