Question

The cubic polynomial with leading coefficient unity all whose roots are $3$ units less than the roots of the equation $x^3-3x^2-4x+2=0$ is denoted as $f\left(x\right)$, then $f^{\prime }\left(x\right)$ is equal to:

• A. $3x^2-12x+5$
• B. $3x^2+12x+5$
• C. $3x^2+12x-5$
• D. $3x^2-12x-5$

Answer 1

The correct option is B $3x^2+12x+5$

Let $\pi :x^3-3x^2-4x+12=0$

Sum of roots taken one at a time of $\pi$

$\Rightarrow \alpha +\beta +\gamma =3$

of $f\left(x\right)=(\alpha -3)+(\beta -3)+(\gamma -3)=3-9=-6$

$\Rightarrow \frac{-b}{a}=-6\Rightarrow b=6\{a=1\}$

Sum of roots taken two at a time of $\pi$

$\alpha \beta +\beta \gamma +\gamma \alpha =-4$

of $f\left(x\right)=(\alpha -3)(\beta -3)+--$

$=\alpha \beta -3(\alpha +\beta )+9+--$

$=(\alpha \beta +\beta \gamma +\gamma \alpha )-3(\alpha +\beta +\beta +\gamma +\gamma +\alpha )+27$

$=-4-3\left(6\right)+27=5$

$\Rightarrow \frac{c}{a}=5\Rightarrow c=5$

$\Rightarrow f\left(x\right)$ is of from $x^3+6x^2+5x+d=0$

$f^{\prime }\left(x\right)=3x^2+12x+5\Rightarrow \left(B\right)$