AMC 10 · 2024 · #15
Grade 11 algebraPick an answer.
The obvious move is to solve the cubic and plug the roots in. Tool #16 (Change Focus) kills that idea early: a quick rational-root test shows the roots are not nice, so the answer cannot come from the roots individually — it has to come from the polynomial as a whole. Tool #13 (Convert to Algebra) supplies the bridge, turning the sentence "p, q, r are the roots" into the identity f(x) = (x-p)(x-q)(x-r), which reports the product of any input's distances to all three roots at once. Then Tool #15 (Organize Information in More Ways) does the real work in two passes: first it rewrites each p²+4 as (p-2i)(p+2i), which is only possible once i is allowed, and then it re-sorts the resulting six factors into two piles that each match the factored polynomial. The catch is that each pile sits backwards relative to f, so three minus signs come out of each pile — the sign bookkeeping is the entire difficulty here. Tool #9 (Solve an Easier Related Problem) audits that sign on a cubic whose roots are visible, before the real numbers get substituted.
Check whether the roots are findable
Every rational candidate fails, so do not chase roots.
A root would show up as a zero in this list; nothing is zero, so chasing the roots is a dead end and the answer must come from the polynomial itself.
11.A-APR.B.2Change Focus Count The ComplementTurn roots into a factored polynomial
The roots are just the factors of f.
The factored form converts one input into the product of its three gaps to the roots, which is exactly the kind of product the question is asking for.
The factored form turns one input into the product of its three gaps to the roots.
▸ Why?
A polynomial vanishing at a root carries that root's linear factor, so the factoring is exact.
▸ Why?
The coefficients record the symmetric combinations of the roots, which is what the products will assemble.
Split using i
Square-plus-four becomes two factors using 2i.
Allowing i turns a sum of squares, which cannot be broken up over the reals, into a difference of squares, which always factors.
11.N-CN.A.1Organize Information In More WaysRegroup six factors into two
The product folds into f at 2i times f at -2i.
Each pile of three reversed subtractions carries one minus sign, and two piles of -1 multiply back to +1.
9.A-SSE.A.2Organize Information In More WaysAudit the sign on a solvable cubic
A toy cubic confirms the sign is right.
An identity claimed for every monic cubic can be checked on the easiest monic cubic there is, and a sign error would show up immediately.
11.A-APR.C.4Solve An Easier Related ProblemEvaluate at 2i and -2i
The two values are conjugates.
Real coefficients send conjugate inputs to conjugate outputs, so the second evaluation is a free mirror image of the first.
11.N-CN.A.2Organize Information In More WaysMultiply the conjugate pair
Their product is the real number 125.
A complex number times its conjugate is the sum of the squares of its two parts, so the imaginary mess is guaranteed to disappear.
11.N-CN.A.3Organize Information In More WaysWhen a question is built out of a polynomial's roots but the roots are ugly, stop hunting for them: rewrite the request as the polynomial evaluated at a couple of well-chosen inputs, and count your minus signs on the way.
- Check whether the roots are findable
- Turn "roots" into a factored polynomial
- Split p² + 4 using i
- Regroup six factors and pay the signs
- Audit the sign on a cubic you can solve
- Evaluate f at 2i and -2i
- Multiply the conjugate pair