In an equivalent-expression question, two forms must give the same output for every input. If the terms share a greatest common factor, use it to choose a product to test first. Graph the original and that product in Desmos, replacing the problem’s letter with both times. A shared point isn’t enough: the graphs must overlap.
Hints
- Hint 1
To find the greatest common factor, take the largest number dividing every coefficient and the smallest power of appearing in every term. What outside factor do those two pieces make?
- Hint 2
A graph shows the output for each input. Change every to in both expressions, then compare their whole curves. Two expressions can agree at one input without being equivalent.
Step-by-step
Approach 1: Test a likely product in Desmos
- Step 1
Graph the original expression
Type in Desmos. Replace every with so is the input and is the output. Leave this graph in place to compare it with a product.
- Step 2
Pick a product to test first
The greatest common factor is : divides all three coefficients, and the smallest power in every term is . So test the product with outside first. A smaller outside factor could still work, but its inside would have to match too.
- Step 3
Check the whole curve
Type below the first line, changing every to . Desmos draws the same curve; hiding and showing the new line doesn’t move it. Matching at one input isn’t enough; equivalent expressions match throughout. The matching expression is . Choice D.
Approach 2: Factor by hand to see why it matches
- Step 1
Divide each term by the common factor
Use the found above. Divide each term by it; for powers of , subtract the exponents:
- Step 2
Write the equal product
Put those three results in parentheses after :
This factorization gives the original expression for every , so it confirms the matching product. Choice D.