Which of the following expressions is equivalent to
?
For expression-equivalence questions, first look for common factors and opportunities to use the distributive property in reverse, as in . Factor out what repeats, simplify what remains inside the parentheses carefully (watching signs when subtracting a binomial), and then factor again if possible to match a choice. Only expand everything if factoring is not obvious, and if you are unsure, you can quickly test a couple of simple x-values (like or ) in both the original expression and each option to see which one always matches.
Hints
Look for a common factor
Both terms in share the same binomial. What factor appears in each term?
Use the distributive property in reverse
Try rewriting the expression as one common factor multiplied by a bracket: something like . What should go inside the bracket?
Simplify inside, then factor again
After you write , carefully simplify the part inside the brackets, then see if that new binomial can be factored further to match one of the choices.
Desmos Guide
Enter the original expression
In the first Desmos line, type the original expression as a function, for example: y1 = (x - 2)(x + 5) - (x - 2)(3x - 1).
Enter each answer choice as a separate function
On new lines, enter each option as a function: y2 = (x - 2)(x - 3), y3 = 2(x - 2)(x - 3), y4 = (x - 2)(x + 3), and y5 = -2(x - 2)(x - 3).
Compare the graphs to find the equivalent expression
Look at the graphs: the correct choice will be the one whose graph lies exactly on top of the graph of y1 for all visible x-values (they should be indistinguishable). That option is the expression equivalent to the original.
Step-by-step Explanation
Factor out the common binomial
Notice that both terms contain the factor :
Use the distributive property in reverse (factoring):
Simplify inside the brackets
Now simplify the expression inside the square brackets:
Combine like terms:
- Combine and : .
- Combine and : .
So the bracket becomes , and the expression is
Factor the new binomial
Factor :
- Both terms have a common factor of .
Now substitute this back into the expression:
Rewrite as a product of three factors and match a choice
Use associativity (you can regroup multiplication) to write
So the original expression is equivalent to , which matches answer choice D.