Question 188·Medium·Equivalent Expressions

After you combine the two fractions, carefully distribute the minus sign through the second numerator and simplify the resulting expression in the numerator.

Once you have a single fraction, see if the numerator has a common factor you can pull out, and then look for a special factoring pattern (like a difference of cubes) that introduces a factor of $x-1$ to cancel with the denominator.

After canceling any common factors with the denominator (using $x \ne 1$), multiply out the remaining factors to get a polynomial, then match it to one of the answer choices.

In the first line of Desmos, type the original expression exactly as given: ((x^5 - 1)/(x - 1)) - ((x^2 - 1)/(x - 1)). You can either graph it or click the table icon to see numerical values for different $x$ (avoiding $x = 1$).

On separate lines, enter each answer choice as its own expression: x^4 + x^3 + x, x^3 + x^2 + x, x^4 + x^2 + 1, and x^4 + x^3 + x^2. Either:

• Graph them and see which graph lies exactly on top of the graph of the original expression for all $x \ne 1$, or
• Create tables for each and compare the $y$-values at several $x$-values (like $x=0,2,3$). The option whose values match the original expression for every tested $x$ is the equivalent expression.

Both terms have the same denominator, $x-1$, so you can combine them into a single fraction:

Be careful to subtract the entire numerator $x^2 - 1$, including the $-1$.

Now simplify the numerator of the combined fraction:

So the expression becomes

Factor out the common factor $x^2$ from the numerator:

Next, recognize that $x^3 - 1$ is a difference of cubes:

Substitute this into the expression:

Because $x \ne 1$, the factor $x-1$ in the numerator and denominator can be canceled:

Now expand this product:

• $x^2 \cdot x^2 = x^4$
• $x^2 \cdot x = x^3$
• $x^2 \cdot 1 = x^2$

So the simplified expression is

which corresponds to choice D.