Question

Suppose for fixed real numbers a and b $f\left(x\right)=x^3+a{x}^2+bx+c$ has 3 distinct roots for $c=0$. Then

• A. $f_c\left(x\right)$ has 3 distinct roots for all real c
• B. $f_c\left(x\right)$ has 3 distinct roots for all real $c>0$ or for all real $c<0$ but not for both
• C. $f_c\left(x\right)$ has 3 distinct roots for all real c in $(p,q)$ for some $p<0$ and $q>0$
• D. $f_c\left(x\right)$ need not have 3 distinct roots for any real c other than zero

Answer 1

The correct option is D $f_c\left(x\right)$ need not have 3 distinct roots for any real c other than zero

The point here is to reduce the equation to

$x^2(x-a)+bx-c=0$

Now if b=1, the equation becomes

$(x^2+1)+(x-c)=0$

The above equation has one real and two imaginary roots and hence does not satisfy the condition given in the question (3 real roots)

Therefore b cannot be equal to 1.