Question

The polynomial $P(x)=x^3+a{x}^2+bx+c$ has the property that the mean of its roots, the product of its roots and the sum of its coefficient are all equal. If the $y$-intercept of the graph of $y=P(x)$ is 2 , then 4
Minimum value of $c{x}^2+2ax-b$ is equal to

• A. - 7
• B. - 1
• C. 2
• D. 7

Answer 1

Answer: (A)

$\because$ y-intercept $=c=2$
Mean of its root $=$ product of roots
$\Rightarrow \frac{-a}{3}=-c\Rightarrow a=6$
$\because 1+a+b+c=\text{ sum of coefficient }=-2$
$\therefore b=-11$
$\therefore c{x}^2+2ax-b=2x^2+12x+11=2(x+3{)}^2-7$
$\therefore \text{ Minimum value }=-7$