AMC 10 · 2017 · #21
Grade 9 algebraPick an answer.
The rule looks open-ended — the candidate roots range over all the integers, and there is no obvious reason the set should ever stop growing. So the question splits into two opposite halves, and both must be settled. The ceiling half is where Tool #3 (Eliminate Possibilities) does the heavy lifting: a single divisibility fact about polynomial roots shows that a newcomer must divide a number already present, which fences the whole process inside a short, explicit list of integers. That converts an infinite search into a finite one. The attainment half then runs Tool #11 (Work Backwards): for each number on that short list, start from the root you want and design a polynomial that has it, using only coefficients already in the set. Tool #2 (Make a Systematic List) keeps the two halves honest by recording the set after each addition, since a construction is legal only if its coefficients were present at the time it was used. When the ceiling and the constructions land on the same set, the count is proved, not guessed.
A root must divide a coefficient
A whole root always divides a coefficient.
Pull out every factor of x you can; whatever constant is left behind is a number that x has to divide.
Pull out every factor of the variable you can; whatever constant is left is a number the root has to divide.
▸ Why?
A polynomial vanishes at a root, so the leftover constant must be cancelled by the terms carrying that root.
▸ Why?
That cancellation is only possible when the division leaves no remainder, which is exactly divisibility.
The process is trapped inside the divisors of 10
That traps everything inside nine possible values.
A divisor of a divisor of 10 still divides 10, so the process can never break out of that nine-number pen.
4.OA.B.4Eliminate PossibilitiesBuild the two signs and -10
Simple polynomials build the first few.
At x = 1 a polynomial is just the sum of its coefficients, so making that sum zero is the whole job.
9.A-CED.A.1Work BackwardsReach 2, and the rest follows
Reaching one key value unlocks the rest.
Once 1 and a number m are both in, x + m hands you -m for free, so members arrive in ± pairs.
6.EE.A.2Work BackwardsCheck the order, not just the list
The building order also has to work.
The rule is a ladder: each new number can only be built out of rungs already climbed.
6.EE.B.5Make A Systematic ListNothing more can be added
The final set has 9 members, choice (D).
The set has swallowed every divisor of every one of its members, so the rule has nothing left to make.
7.NS.A.2Eliminate PossibilitiesFind the ceiling first — a new number always has to divide a number already in the set, so nothing can escape the divisors of 10 — then build all nine of those numbers with real polynomials and stop.
- A root must divide a coefficient
- The process is trapped inside the divisors of 10
- Build the two signs and -10
- Reach 2, and the rest follows
- Check the order, not just the list
- Nothing more can be added