Question

Let $P\left(x\right)=x^2+bx+c,$ where $b$ and $c$ are integers. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5,$ then the value of $P\left(3\right)$ is

Answer 1

Let $x=\alpha$ be the common root of both the equations.
Then, ${\alpha }^4+6{\alpha }^2+25=0$
$\Rightarrow {\alpha }^4=-6{\alpha }^2-25$
Replace ${\alpha }^4$ value in second equation,
$3(-25-6{\alpha }^2)+4{\alpha }^2+28\alpha +5=0$
$\Rightarrow {\alpha }^2-2\alpha +5=0$
On comparing with $x^2+bx+c,$
$b=-2,c=5$
$\therefore P\left(x\right)=x^2-2x+5$
$\Rightarrow P\left(3\right)=8$