Question

The polynomial $P ( x )= x ^3+ ax ^2+ bx + c =0$ 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 $y=P(x)$ is 2 , then find the value of $b+P(2)$.

Answer 1

$\Theta y$-intercept $\Rightarrow P(0)=c=2$
$\therefore \text { product of roots }=- c =-2 $
$\text { sum of roots }=- a$
$\therefore \text { Arithmetic mean }=\frac{- a }{3}=-2 \Rightarrow a =6 $
$\Theta \text { sum of coefficient }=1+ a + b + c =-2 $
$\Rightarrow 1+6+ b +2=-2$
$\Rightarrow b =-11 $
$\therefore P ( x )= x ^3+6 x ^2-11 x +2 $
$\therefore P (2)=8+24-22+2=12 $
$P (2)+ b =1$