If two remainders are related can we find $a$?

If two remainders are related can we find $a$?

Question

The remainder when $x^4 + 3x^2 - 2x + 2$ is divided by $x + a$ is the square of the remainder when $x^2 - 3$ is divided by $x + a$. Calculate the possible values of $a$.

Solution

The remainder theorem tells us that if a polynomial $p(x)$ is divided by $x-a$, the remainder is $p(a)$.

So on dividing $x^2 - 3$ by $x + a$, we have a remainder of $a^2-3$.

We also have that if $f(x) = x^4 + 3x^2 - 2x + 2$ is divided by $x + a$, the remainder is $f(-a)=a^4 + 3a^2 + 2a + 2$.

Thus the question tells us that $(a^2-3)^2 = a^4 + 3a^2 + 2a + 2 \implies 9a^2+2a-7=0.$

Factorising, we have that $(a+1)(9a-7) = 0$, and so the possible values for $a$ are $-1$ and $\dfrac{7}{9}$.