AMC 10 · 2002 · #25
Grade 9 algebra
Pick an answer.
There is no formula to compute here, and P is never given, so the only way in is to find something that is true of P and Q for every possible P and then check the five pictures against it. Averaging destroys almost everything about a list of numbers, so the search is for what it protects. It protects the total: a list of k numbers with mean m adds up to km, which is the same total as before. A coefficient total is not obviously a picture fact, but substituting x = 1 turns any polynomial into the sum of its coefficients, so the protected total becomes a single shared point on both graphs. That is the whole problem: the two curves are forced to meet at x = 1. A second, independent test comes from writing Q out. Every nonzero coefficient of Q is the same number m, so Q is m times a bare sum of powers, and for positive x that sum is positive; this pins down the sign and the direction of Q on the whole right half of every legal picture. Finally, since the question asks what could be, elimination alone is not a finish: the surviving panel needs an explicit P built for it.
Write down what Q actually is
Q keeps P's exponents with one shared coefficient: Q(x) = m(x^e₁ + … + x^e_k), so both share a degree.
When every coefficient becomes the same number, that number factors out and what is left is a plain sum of powers with no coefficients at all.
9.A-SSE.A.1Introduce A VariableThe mean protects the total
A mean preserves exactly one thing, the total: k copies of m add to the same sum as P's coefficients.
Levelling a list of numbers to their average is just spreading the same total evenly, so the total never moves.
Replacing every coefficient by their average spreads the same total evenly, so the total never moves.
▸ Why?
The average is the total shared out equally, so multiplying it back by the count returns the very same total.
▸ Why?
The sum of the coefficients is what the separate coefficients come to together, so levelling them rearranges without adding or losing any.
Turn the total into a point on both graphs
Substituting x = 1 turns that total into a shared point, forcing P(1) = Q(1) for every P.
Plugging in x=1 turns a polynomial into the sum of its coefficients, which is exactly the quantity the mean keeps fixed.
9.F-IF.A.2Change Focus Count The ComplementTest the panels at x = 1
Checking where the curves meet, four panels fail to cross at x = 1 and are impossible.
One forced point is enough to convict: if the curves do not meet where they must, no polynomial can be behind the picture.
9.F-IF.B.4Eliminate PossibilitiesConfirm with a second test that never looks at x = 1
A second test confirms it: for positive x, Q keeps m's sign and moves in one steady direction.
With one repeated coefficient there is nothing left to cause cancellation, so on the positive side Q can only march steadily in a single direction.
9.A-SSE.A.2Eliminate PossibilitiesBuild a polynomial for the panel that is left
The witness P(x) = 2x⁴-3x²-3x-4 gives m = -2 and reproduces the survivor, so the answer is (B).
A single explicit polynomial turns "nothing rules this panel out" into "here it is".
9.A-SSE.B.3Guess And CheckAveraging a list of numbers never changes their total, and plugging in x=1 turns a polynomial into the total of its coefficients — so the two curves have no choice but to meet at x=1.
- Write down what Q actually is
- The mean protects the total
- Turn the total into a point on both graphs
- Test the panels at x = 1
- Confirm with a second test that never looks at x = 1
- Build a polynomial for the panel that is left