HMMT-Nov 2019 guts, Problem 1

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: 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 $-7 \leq 4 a \leq 7$, 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. $\fbox{2}$.

Answer: 2