Polynomial $P(x)=x^4+ax^3+2bx^2+cx-b+k$ with real roots

Question

For which largest $k$ the polynomial $P(x)=x^4+ax^3+2bx^2+cx-b+k$ have real roots for arbitrary $a,b$ and $c$?

This is own problem. I can show that there are real roots for $k=-\frac{1}{4}$: $$P\left(\dfrac1{\sqrt2}\right)+P\left(-\dfrac1{\sqrt2}\right)=0.$$

Answer 1

It appears that for $P(x)=x^4+0x^3+2bx^2+0x-b+k$,

• in case $b$ is positive, your $k$ can be as high as $b$, which grows ad infinitum for bigger and bigger values of $b$.

• in case $b$ is negative, your $k$ can be as high as $[b\times (b+1)]$, which grows ad infinitum for smaller and smaller values of $b$.