Question

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1,$ the remainder is $x + 2,$ and when $P(x)$ is divided by the polynomial $x^2 + 1,$ the remainder is $2x + 1.$ There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

$10$

$13$

$19$

$20$

$23$

Answer: E

Concepts:polynomial Chinese Remainder Theorem

Difficulty rating: 2020

Solution:

No linear polynomial can have the two different remainders. If $P(x)=ax^2+bx+c$ were quadratic, comparing its remainders would give both $b-a=1,\ c-a=2$ and $b=2,\ c-a=1,$ a contradiction. Thus the least possible degree is $3.$ Write $P(x) = (x + 2)$ $+ (x^2 + x + 1)(px + q),$ which has remainder $x + 2$ upon division by $x^2 + x + 1.$

Reducing modulo $x^2 + 1$ (so $x^2 \equiv -1$ ) gives remainder $(q + 1)x + (2 - p).$ Setting this equal to $2x + 1$ gives $q = 1$ and $p = 1.$

Then $P(x) = x^3 + 2x^2 + 3x + 3,$ and the sum of the squares of the coefficients is $1 + 4 + 9 + 9 = 23.$

Thus, the correct answer is E.

2022 AMC 12B Problem 20

Answer: E

Difficulty rating: 2020

Solution:

Problem 20 in Other Years