AMC 10 · 2016 · #24
Grade 11 algebraPick an answer.
The question is a minimum question, so the Extreme Principle sets the shape of the work: find a lower bound for a, then show some polynomial actually reaches it. To get a bound, stop looking at the polynomial and look at its roots instead. Naming the roots r, s, t turns the coefficients into Vieta's relations, and the problem's quirk (same a in two places) says sum of roots = product of roots. That is exactly the setup where the AM-GM inequality collapses everything to a single inequality in a. Three subproblems fall out: prove the roots are positive so AM-GM is legal, get the bound, and check the bound is attained.
Turn coefficients into root facts
The coefficients become facts about the roots.
Matching coefficients converts a statement about a polynomial into three plain equations about its roots.
11.A-APR.C.4Introduce A VariableProve all three roots are positive
All three roots must be positive.
A positive product would allow two negative roots, but two negative roots drag the middle coefficient below zero.
9.A-CED.A.3Organize Information In More WaysSqueeze a with AM-GM
A mean inequality then bounds the coefficient.
Because sum and product are the same number here, AM-GM stops being about the roots and becomes an inequality about a alone.
Because the sum and the product are the same number here, the balance between the roots becomes an inequality about one coefficient.
▸ Why?
For a fixed total the parts multiply to the most when they are equal, and pulling them apart only costs.
▸ Why?
A polynomial's coefficients already record the sum and the product of its roots, so both sides are readable.
Find what equality forces
Equality forces all three roots equal.
A lower bound only matters if some case actually sits on it, and AM-GM sits on its bound only when everything is equal.
11.N-RN.A.2Extreme PrincipleCheck that the bound is reached
That cubic really exists.
An inequality only names a floor; you still have to show one example standing on it.
11.A-APR.B.3Identify SubproblemsRead off the forced b
Reading the other coefficient gives 9, choice (C).
Once the roots are pinned, every coefficient is pinned with them.
11.A-APR.C.4Introduce A VariableWhen the roots are forced to have the same sum as product, AM-GM squeezes that number from below, and the tightest case is always the one where all the roots are equal.
- Turn coefficients into root facts
- Prove all three roots are positive
- Squeeze a with AM-GM
- Find what equality forces
- Check that the bound is reached
- Read off the forced b