Question

If $f\left(x\right)$ is a polynomial of degree $4$ with leading coefficient $4$ and $f\left(1\right)=2,f\left(2\right)=8,f\left(3\right)=18,f\left(5\right)=50$, then $f\left(4\right)$ is

Answer 1

$f\left(1\right)=2,f\left(2\right)=8,f\left(3\right)=18,f\left(5\right)=50$
By given values, we assume
$f\left(x\right)=2x^2$ for $x=1,2,3,5$
Let $g\left(x\right)=f\left(x\right)-2x^2$, then
$x=1,2,3,5$ are roots of $g\left(x\right)$
As degree of $f\left(x\right)$ is $4$, so degree of $g\left(x\right)$ is also $4$
Therefore,
$g\left(x\right)=a(x-1)(x-2)(x-3)(x-5)\Rightarrow f\left(x\right)=a(x-1)(x-2)(x-3)(x-5)+2x^2$
As the leading coefficient of $f\left(x\right)$ is $4$, so
$f\left(x\right)=4(x-1)(x-2)(x-3)(x-5)+2x^2\Rightarrow f\left(4\right)=4\left(3\right)\left(2\right)\left(1\right)(-1)+32\therefore f\left(4\right)=8$