AMC 10 · 2010 · #21
Grade 11 algebraPick an answer.
Tool #16 (Change Focus): comparing two graphs is awkward, so I stop looking at the curve and the line separately and look only at their difference D(x); the whole hypothesis becomes one sentence about a single polynomial. Tool #14 (Extreme Principle): every meeting point is a place where D hits its floor value 0 without ever going lower, and that boundary behaviour is what forces each root to be a double root — three doubles then use up all six degrees exactly, so D must be the square of a cubic. Tool #4 (Introduce a Variable): I name that cubic x³-ux²+vx-w with unknown u,v,w, square it once in general letters, and match coefficients; the unknowns a,b,c only touch the bottom three coefficients, so the top four determine u,v,w on their own. Tool #3 (Eliminate Possibilities): at the end I test the answer choices directly in the cubic, which both checks the factoring and kills the other four choices.
Compare by subtracting
Subtracting turns the comparison into one sign question.
One graph sitting above another is the same fact as one polynomial staying at or above zero.
11.A-REI.D.11Change Focus Count The ComplementA touching zero is a double zero
Every touching root must be double.
A graph forbidden to dip below zero can only touch the axis, never cross it, and a touch always costs two roots.
A graph forbidden to dip below zero can only touch the axis, never cross it, and a touch costs two roots.
▸ Why?
A polynomial vanishes where a factor vanishes, and crossing would mean that factor appears an odd number of times.
▸ Why?
An odd count of a factor makes the value change sign there, which the non-negativity forbids.
Three double zeros fill degree 6
Three double roots fill the degree exactly.
Six degrees and three touches at two each is a perfect fit, so nothing else can be hiding in the factorisation.
11.N-CN.C.9Extreme PrincipleSquare a cubic in general letters
So the difference is a cubic squared.
One general expansion replaces endless trial cubics.
11.A-APR.C.4Introduce A VariableMatch the four known coefficients
The four known coefficients fix that cubic.
The unknown numbers all sit in the bottom half of the polynomial, so the top half is already fully known.
9.A-SSE.A.2Introduce A VariableFactor the cubic
It factors into three whole-number roots.
Once the cubic is pinned down, the meeting points are simply its roots.
11.A-APR.B.3Introduce A VariableConfirm the picture is real, then read off the answer
The largest of them is 4, choice (A).
Building the actual example upgrades 'these are the only candidates' into 'these are the answer'.
9.A-SSE.B.3Eliminate PossibilitiesA polynomial that is never negative can only touch zero, never cross it, so every touch is a double root - and three touches use up a degree-6 polynomial exactly, forcing it to be the square of a cubic.
- Compare by subtracting
- A touching zero is a double zero
- Three double zeros fill degree 6
- Square a cubic in general letters
- Match the four known coefficients
- Factor the cubic
- Confirm the picture is real, then read off the answer