Question

A polynomial $p\left(x\right)$ of second degree which will vanish at $x=3$ and $x=5$ and will have value of $4$ when $x=2$, is in the form of $a{x}^2+bx+c$. The value of $[a+b+c]$ is
(where $[\cdot ]$ deontes the greatest integer function).

Answer 1

$p\left(x\right)$ vanishes at $x=3$ and $x=5$
$\therefore p\left(x\right)=a(x-3)(x-5)$
Given, $p\left(2\right)=4$
$\Rightarrow a(2-3)(2-5)=4$
$\Rightarrow a=\frac{4}{3}$
$\therefore p\left(x\right)=\frac{4}{3}(x-3)(x-5)$
$=\frac{4}{3}(x^2-8x+15)$
$=\frac{4x^2}{3}-\frac{32x}{3}+20$
on comparing with $p\left(x\right)=a{x}^2+bx+c$
$[a+b+c]=[\frac{4}{3}-\frac{32}{3}+20]=\left[\frac{32}{3}\right]=10$