Question: For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$?

Question

For what value of $c$ will $P(x)= -2x^3+cx^2-5x+2$ have the same remainder when devided by $x+1$ or $x-2$?

No idea where to start with this one. Can't use the remainder theorem to find the remainder because of the leading coefficient and can't find the value of c when I have no known remainder. Be nice if the examples in my schools lessons actually included the same kind of questions that are on the tests but I guess that would make to much sense for school.

Answer 1

Hint: use polynomial long division to divide $P(x)$ by $x+1$, and find the remainder term in terms of $c$. Then, divide $P(x)$ by $x-2$, and also find the remainder term in terms of $c$. Set the two remainder terms equal and solve...

Answer 2

Remainder theorem: When $P(x)$ is divided for $(x-k)$, the remainder is $P(k)$

Dividing by $(x+1)$ the remainder is $P(-1)$ and dividing by $(x-2)$ is $P(2)$

So we need to set $$P(-1)=P(2)\to 9+c=-24 + 4 c\to c=11$$