Question

The equation $P\left(x\right)=x^3+a{x}^2+bx+c=0$ has the property that the arithmetic mean of its roots, the product of its roots and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=P\left(x\right)$ is $2,$ then the value of $b+P\left(2\right)$ is

Answer 1

$y$-intercept is $2$
$\Rightarrow P\left(0\right)=c=2$

Given,
Product of roots $=$ arithmetic mean of roots $=$ sum of coefficients
Product of roots $=-c=-2$
Sum of roots $=-a$
Arithmetic mean $=\frac{-a}{3}=-2\Rightarrow a=6$

Sum of coefficients $=1+a+b+c=-2$
$\Rightarrow 1+6+b+2=-2\Rightarrow b=-11$

$\therefore P\left(x\right)=x^3+6x^2-11x+2$
$\Rightarrow P\left(2\right)=8+24-22+2=12$
$\Rightarrow P\left(2\right)+b=1$