Technique Encyclopedia
Exploiting Answer Choices
What it is
On a multiple-choice test, the answer choices are extra information. Sometimes it's faster to test each option directly against the problem's conditions than to solve for the answer from scratch. Other times, a quick property check — parity, sign, magnitude — eliminates most of the choices instantly, leaving little or no work to identify the correct one.
Signals that suggest using it
- Solving the problem directly looks slow, but checking whether a given value satisfies the condition looks fast.
- The answer choices have an obvious shared property (all integers, a clear sign, a specific range) that the true answer must also satisfy.
- An equation is much easier to verify at a specific value than to solve symbolically.
When it's effective
Best when direct verification of a candidate answer is dramatically faster than solving the problem the intended way — testing a cubic equation's given roots, for instance, beats factoring it from scratch.
When it's not effective
Doesn't apply to free-response formats, and even in multiple-choice settings it's no help when verifying each choice takes about as much work as solving the problem directly.
Simple example
Problem
Which value of x satisfies 2x + 3 = 11? (A) 2 (B) 3 (C) 4 (D) 5
Solution
Testing (C): 2(4) + 3 = 11. ✓ The answer is (C).
AMC-style example
Problem
The equation x³ − 6x² + 11x − 6 = 0 has three integer roots. Which of the following is the largest root? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Solution
Testing each option by direct substitution is much faster than factoring the cubic: (D) gives 64 − 96 + 44 − 6 = 6 ≠ 0, so 4 is not a root. (C) gives 27 − 54 + 33 − 6 = 0. ✓ Since 4 and 5 both fail, (C) 3 is the largest actual root.