Question

Let $P ( x )= x ^4+ ax ^3+ bx ^2+ cx +320$ be a polynomial where $a , b , c \in R$. If $| P (4)|+\left| P ^{\prime}(4)\right|+\left| P ^{\prime \prime}(4)\right| \leq 0$, then find the value of $11 a - b - c$.

Answer 1

$\Theta | P (4)|+\left| P ^{\prime}(4)\right|+\left| P ^{\prime \prime}(4)\right| \leq 0$
${\Rightarrow}P(4)= P ^{\prime}(4)= P ^{\prime \prime}(4)=0$
$\therefore x =4$ will be a repeated root of $P ( x )=0$ repeating thrice.
Let root of $P ( x )=0$ be $x _1, x _2, x _3, x _4$
$\therefore x _1= x _2= x _3=4$
P Product of roots $= x _1 x _2 x _3 x _4=320$
$\Rightarrow x _4=5$
$\therefore $ Sum of roots $=4+4+4+5=- a \Rightarrow a =-17$
Product of roots taken two at a time
$=x_1 x_2+x_1 x_3+x_1 x_4+x_2 x_3+x_2 x_4+x_3 x_4=b $
$\Rightarrow b=16+16+20+16+20+20=108$
and product of roots taken three at a time
$ = x _1 x _2 x _3+ x _1 x _2 x _4+ x _1 x _3 x _4+ x _2 x _3 x _4=- c$
$\Rightarrow - c =64+80+80+80 \Rightarrow c =-284 $
$\therefore 11 a - b - c =-187-108+304=$