Question

If $p\left(x\right)=a{x}^2+bx+c$ has more than 2 roots then which of the following is correct:

• A. $a=0$
• B. $b=0$
• C. $b=0,c=0$
• D. None of these

Answer 1

The correct option is D None of these

Let $p\left(x\right)$ has three distinct roots $x_1,x_2,x_3$

As $x_1,x_2$ are roots of $p\left(x\right)$, we can

write $p\left(x\right)$ as:

$p\left(x\right)=(x-x_1)(x-x_2)g\left(x\right)$

Since $p\left(x\right)$ has degree $2,g\left(x\right)$ must be

a constant k.

Further, we have $p\left(x_3\right)=0$

$\Rightarrow k(x_3-x_1)(x_3-x_2)=0$

Since we assumed $x_1,x_2,x_3$ to be

distinct (If any two would be same,

roots won't be 3 anymore), k must be

equal to zero. Hence, $p\left(x\right)$ is zero

for any value of x. Hence, the only

possibility for this to happen is if

$a=b=c=0$.

No Option matches correctly.