Which expression is equivalent to
For rational-expression equivalence questions, first factor all numerators and denominators completely and cancel any common factors before combining fractions. Then get a common denominator, combine the numerators carefully (watching minus signs), and simplify the resulting numerator and denominator as far as possible. If you’re unsure between a couple of options, quickly test a simple value of (that doesn’t make any denominator zero) to see which expression matches the original.
Hints
Start by factoring
Look at the polynomials and . Can you factor them? One is a classic difference of squares.
Simplify the first fraction
Once you factor the numerator and denominator of the first fraction, check whether a common factor appears in both so you can cancel it before doing any subtraction.
Use a common denominator
After simplifying the first fraction, notice what the denominators of the two fractions are. How can you combine them into a single fraction so you only subtract the numerators?
Be careful with the minus sign
When you subtract the second fraction, make sure the minus sign is distributed across both terms in the numerator , not just the first term.
Desmos Guide
Enter the original expression
In Desmos, type the original expression as a function, for example:
y1 = (3x^2 - 2x - 1)/(x^2 - 1) - (2x + 3)/(x + 1)
This will graph the given expression.
Graph each answer choice
On new lines, enter each choice as its own function:
y2 = (x - 2)/(x + 1)y3 = (x - 2)/(x - 1)y4 = (x + 2)/(x + 1)y5 = (x - 2)/(x^2 - 1)
You will see four additional graphs.
Compare graphs or use tables
Zoom out or adjust the viewing window so you can clearly see all graphs. The correct choice will have a graph that lies exactly on top of the graph of y1 for all where both are defined (there may be holes at ). You can also click the gear icon for each function and add a table to compare -values of y1 and each candidate at several -values; the equivalent expression will match y1's values every time.
Step-by-step Explanation
Factor the polynomials
First, factor the denominator as a difference of squares:
Now factor the numerator . You want two numbers that multiply to and add to . Those numbers are and .
Rewrite and factor by grouping:
So the first fraction becomes
Cancel the common factor in the first fraction
In
the factor appears in both the numerator and the denominator, so for you can cancel it:
Now rewrite the entire expression using this simplification:
Combine the fractions using the common denominator
Both fractions now have the same denominator , so you can combine them into a single fraction by subtracting the numerators:
Be sure to keep the parentheses so the minus sign applies to the whole second numerator.
Simplify the numerator to get the final expression
Simplify the numerator:
So the entire expression simplifies to
which matches answer choice D.