Question

$f\left(x\right)$ is a polynomial of degree 4 with real coefficients such that $f\left(x\right)=0$ is satisfied by $x=1,2,3$ only, then $f^{\prime }\left(1\right).f^{\prime }\left(2\right).f^{\prime }\left(3\right)$ is equal to

• A. $0$
• B. $2$
• C. $-1$
• D. none of these

Answer 1

The correct option is A $0$

Since, $x=1,2,3$ only satisfies $f\left(x\right)=0$
So, $x=1,2,3$ are the roots of $f\left(x\right).$
Since, $f\left(x\right)$ is a polynomial of degree 4 .So one root is repeated.
$\Rightarrow f^{\prime }\left(1\right)=0$ or $f^{\prime }\left(2\right)=0$ $f^{\prime }\left(3\right)=0$
So, $f^{\prime }\left(1\right).f^{\prime }\left(2\right).f^{\prime }\left(3\right)=0$