Question

Let $P ( x )$ be the polynomial $x ^3+ ax ^2+ bx + c$, where $a , b , c \in R$. If $P (-3)= P (+2)=0$ and $P^{\prime}(-3)<0$, which of the following is a possible value of ' $c^{\prime}$ ?

• A. -27
• B. -18
• C. -6
• D. -3

Answer 1

Answer: (A)

$ P ( x )= x ^3+ ax ^2+ bx + c$
$P ^{\prime}( x )=3 x ^2+2 ax + b$
$P (-3)=0 \Rightarrow -27+9 a -3 b + c =0$ ....(1)
$P (2)=0 \Rightarrow 8+4 a +2 b + c =0$ ....(2)
hence $(1)-(2)$ gives
$5 a -5 b -35=0$
$a - b =7$ ....(3)
$P^{\prime}(-3)=27-6 a+b<0 \Rightarrow 27-6(a-b)-5 b<0 \Rightarrow 27-42-5 b<0$
$\Rightarrow -15-5 b <0 \Rightarrow 3+ b >0 \Rightarrow b >-3$, hence $a -7= b >-3$
$a-7 >-3 [$ from $(3)]$
$a >4$....(4)
$\therefore $ from (2)
$8+16-6+c_{\max }=0$
$c_{\max }=-18 \stackrel{\max }{\Rightarrow} c <-18$