For all real numbers x such that x \neq 5, which of the following expressions is equivalen… — GRE General Test

For all real numbers $x$ such that $x \neq 5$ , which of the following expressions is equivalent to $\frac{x^2 - 25}{3x - 15}$ ?

The expression

\frac{x + 5}{3}

is equivalent to the given rational expression for all

x \neq 5

Factoring the numerator as a difference of squares gives

(x - 5)(x + 5)

, and factoring the denominator gives

3(x - 5)

. Dividing out the common factor

(x - 5)

simplifies the expression to

\frac{x + 5}{3}

Factor the numerator using the difference of squares identity.

x^2 - 25 = (x - 5)(x + 5)

The difference of squares

a^2 - b^2

factors into

(a - b)(a + b)

Factor out the greatest common factor from the denominator.

3x - 15 = 3(x - 5)

Both terms in

3x - 15

share a common factor of

3

Divide out the common binomial factor

(x - 5)

from the numerator and denominator.

\frac{(x - 5)(x + 5)}{3(x - 5)} = \frac{x + 5}{3}

Since

x \neq 5

, the factor

(x - 5)

is non-zero and can be cancelled.

Simplifying rational algebraic expressions by factoring difference of squares and common linear factors.

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