Question

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:

$3$

$5$

$6$

$7$

$9$

Answer: B

Concepts:polynomial casework

Difficulty rating: 1730

Small Hint:

Since $n+a_0\le3$ and $a_0\ge1,$ consider only $n=0,1,2$

Big Hint:

For each degree, count the integer coefficient tuples with the required sum of absolute values

Solution:

We count by degree. If $n=0,$ then $a_0=3,$ giving one polynomial. If $n=1,$ then $a_0+|a_1|=2.$ This gives $a_0=2,a_1=0$ or $a_0=1,a_1=\pm1,$ for three polynomials. If $n=2,$ then $a_0=1$ and $a_1=a_2=0,$ giving one more. No higher degree is possible. The total is $1+3+1=5.$

Thus, the correct answer is B.

1959 AMC 12 Problem 48

Answer: B

Difficulty rating: 1730

Small Hint:

Big Hint:

Solution:

Problem 48 in Other Years