An equivalent expression gives the same value for every value of its variable. In a multiple-choice product question, setting the variable to gives a quick Desmos test: every term containing it disappears. If only one choice matches, you can select it; otherwise, distribute and combine like terms. One matching input alone does not prove equivalence.
Hints
- Hint 1
Set to find the constant term, the part left when every term containing disappears. What value does the given product have at that input?
- Hint 2
An equivalent expression must agree with the original at every input, including . Compare the choices at that value. Does the test leave only one choice?
Step-by-step
Approach 1: Eliminate choices with Desmos
- Step 1
Find the product's value at zero
Set so the constant term, the part without , is all that remains. Type and in Desmos. It prints , so any equivalent expression must also give at .
- Step 2
Test the choices at zero
Keep those lines and type the four choices beneath them, in order: , , , and . Desmos prints , , , and . Only matches the original. One input can rule out choices but cannot prove a match by itself. Here it rules out every other choice, so is the equivalent expression. Choice A.
Approach 2: Distribute to prove the match
- Step 1
Write all four products
The distributive property says each term in the first parentheses multiplies each term in the second. Write all four products before combining anything:
- Step 2
Multiply each pair
Multiply the four pairs, keeping the negative sign in :
- Step 3
Combine the matching terms
Like terms have the same variable part, so only and combine:
This shows the expressions match for every , not only at . The equivalent expression is . Choice A.