SAT Equivalent Expressions Question 74 | Free SAT Prep

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Question 74·Medium·Equivalent Expressions

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For $x \ne 2$ , which expression is equivalent to

\frac{x^{3} - 8}{x - 2}?

When a rational expression has a higher-degree polynomial over a linear binomial like $x - 2$ , first check if the numerator factors using a special pattern (difference of squares or cubes) that contains the same binomial. Rewrite constants as powers (e.g., $8 = 2^3$ ), apply the correct formula (such as $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ ), and then cancel any common binomial factor, being careful that the excluded value (like $x \ne 2$ ) makes the cancellation valid. As a quick check, multiply your simplified answer by the original denominator to see if you recover the original numerator.

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Rewrite the constant term

Try rewriting $8$ as a power: what simple integer cubed gives $8$ ?

Use a special factoring pattern

Once you see $x^3 - 2^3$ , think of the formula for factoring a difference of cubes $a^3 - b^3$ .

Factor, then simplify the fraction

After factoring the numerator using the difference of cubes pattern, see if any factor in the numerator is the same as the denominator so you can cancel it (keeping in mind $x \ne 2$ ).

Desmos Guide

Enter the original rational expression

Type f(x) = (x^3 - 8)/(x - 2) into Desmos. Notice that the graph will have a hole at $x = 2$ because the denominator is zero there.

Enter each answer choice as separate functions

Type each option as its own function, for example: g(x) = x^2 + 2x + 4, h(x) = x^2 - 4, p(x) = x^2 + 4, and q(x) = x^2 + 2.

Compare the graphs away from x = 2

Look at the graphs of all these functions. The correct choice is the one whose graph lies exactly on top of the graph of $f(x)$ for all $x$ values except at $x = 2$ (where $f$ has a hole). That function is the expression equivalent to the original fraction.

Step-by-step Explanation

Recognize the structure of the numerator

Notice that $8$ can be written as $2^3$ , so the numerator is

x^3 - 8 = x^3 - 2^3,

which is a difference of cubes.

Recall and apply the difference of cubes formula

The difference of cubes formula is

a^3 - b^3 = (a - b)(a^2 + ab + b^2).

Here, $a = x$ and $b = 2$ , so

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

Simplify the rational expression by canceling the common factor

Substitute the factored form of the numerator into the fraction:

\frac{x^3 - 8}{x - 2} = \frac{(x - 2)(x^2 + 2x + 4)}{x - 2}.

Because $x \ne 2$ , the factor $x - 2$ in the numerator and denominator cancels, leaving

\frac{x^3 - 8}{x - 2} = x^2 + 2x + 4.

So the equivalent expression is $x^2 + 2x + 4$ . This matches answer choice A.

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Step-by-step Explanation

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Step-by-step Explanation