HMMT 十一月 2019 · 冲刺赛 · 第 1 题
HMMT November 2019 — Guts Round — Problem 1
题目详情
- [5] 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?
解析
- [5] 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? Proposed by: Carl Schildkraut Answer: 2 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) = ax − 4 a for some nonzero integer a . Since all integer coefficients are between − 7 and 7, inclusive, we require − 7 ≤ 4 a ≤ 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.