Polynomial Functions Question (Determine values of $~k~$ and $~f~$)

Question

The question is

When $~2x^3 + x^2 - 2kx + f~$ is divided by $~x - 1~$, the remainder is $~-4~$, and when it is divided by $~x+2~$, the remainder is $~11~$. Determine the values of $~k~$ and $~f~$.

I know how to solve for $~k~$, you would just sub in the root for $~x~$ and set the equation equal to the remainder but because I also have to solve for $~f~$ this throws me off and I am confused as of what to do.

Answer 1

After you replace the value of $x$ with $1$ and then with $-2$, you will get two equations. The first equation is equal to $-4$ , by Bézout's theorem. Analogously, the second equation is equal to $11$.

That's a linear system of two equations with two unknowns. $$-2k+f=-7 $$ $$ 4k+f=23 $$

Multiply the first equation by 2, and then add that result to the second equation. You can get the value of $f$ that way. After that, obtaining the value of $k$ should be easier.

Answer 2

Hint: If $p(x)=2x^3+x^2-2kx+f$, then $p(1)=-4$ and $p(-2)=11$. So…