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?

$\ 10$

$\ 13$

$\ 19$

$\ 20$

$\ 23$

Solution:

Since $P(x)$ has a remainder of $x+2$ when divided by $x^2+x+1,$ it must be able to be written as $P(x) =$ $(x^2+x+1)Q(x)+x+2$ for some polynomial $Q(x).$ Note that the remainder of $P(x)$ when divided by $x^2+1$ is equal to the remainder when $xQ(x) + x+2.$

If $Q(x)=c$ is constant, this remainder is $(c+1)x+2,$ which can never equal $2x+1$ because the constant terms $2$ and $1$ differ. So $Q$ must have degree at least $1.$

Try $Q(x) = ax+b.$ Then the remainder of $P(x)$ modulo $x^2+1$ equals the remainder of $(ax+b)x+x+2 = ax^2 + (b+1)x+2$ modulo $x^2+1.$ Subtracting $a(x^2+1)$ gives $(b+1)x +(2-a),$ which must equal $2x+1.$ Thus $b+1=2$ and $2-a=1,$ so $a=b=1.$

This means $P(x) =$ $(x+1)(x^2+x+1)+x+2 =$ $x^3+2x^2+3x+3.$ The sum of the squares of the coefficients is $23.$

Thus, the answer is E .

2022 AMC 10B Problem 21

Solution:

Problem 21 in Other Years