Question 48
Given the polynomial $$
\begin{aligned} &a_0x^n+a_1x^{n-1}+\cdots\\ &\qquad{}+a_{n-1}x+a_n, \end{aligned}
$$ where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$’s are integers or zero. Set $$
\begin{aligned} h={}&n+a_0+|a_1|+|a_2|\\ &{}+\cdots+|a_n|. \end{aligned}
$$ [See example $25$ for the meaning of $|x|.$] The number of polynomials with $h=3$ is:
\begin{aligned} &a_0x^n+a_1x^{n-1}+\cdots\\ &\qquad{}+a_{n-1}x+a_n, \end{aligned}
$$ where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$’s are integers or zero. Set $$
\begin{aligned} h={}&n+a_0+|a_1|+|a_2|\\ &{}+\cdots+|a_n|. \end{aligned}
$$ [See example $25$ for the meaning of $|x|.$] The number of polynomials with $h=3$ is: