Question

If $\alpha$ and $\beta$ are the zeroes of the polynomial $f\left(x\right)=x^2-px+q$, then the polynomial having $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ as its zeroes is __________.

• A. $x^2+qx+p$
• B. $x^2-px+q$
• C. $q{x}^2-px+1$
• D. $p{x}^2+qx+1$

Answer 1

Answer: (C)

$q{x}^2-px+1$

If the polynomial $f\left(x\right)=x^2-px+q$ has the roots as $\alpha$ and $\beta$.

Then the equation having the roots as $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is $f\left(\frac{1}{x}\right)$.

$\Rightarrow$ $f\left(\frac{1}{x}\right)=\left(\frac{1}{x^2}\right)-p\left(\frac{1}{x}\right)+q$

$\Rightarrow$ $f\left(\frac{1}{x}\right)=q{x}^2-px+1$.

Therefore the equation having $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ as roots is $q{x}^2-px+1$.