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, 2019

Algebra Difficulty 5.0 AIME, harder Prove it United States

Problem:
A polynomial $P$ with integer coefficients is called tricky if it has $4$ as a root.
A polynomial is called teeny if it has degree at most $1$ and integer coefficients between $-7$ and $7$ , inclusive.
How many nonzero tricky teeny polynomials are there?

Solution

Solution:
If a degree $0$ polynomial has $4$ as a root, then it must be the constant zero polynomial. Thus, we will only consider polynomials of degree $1$ .

If $P$ has degree $1$ , integer coefficients, and $4$ as a root, then it must be of the form $P(x) = a(x-4) = a x - 4 a$ for some nonzero integer $a$ . Since all integer coefficients are between $-7$ and $7$ , inclusive, we require $-7 \leq 4a \leq 7$ , which gives us $a = -1, 1$ . Note that for both values, the coefficient of $x$ is also between $-7$ and $7$ , so there are $2$ tricky teeny polynomials.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.