Question

Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of polynomial $f\left(x\right)=p{x}^2+qx+x,p\ne 0,r\ne 0$

Answer 1

If $\alpha ,\beta$ are the zeroes of polynomial $f\left(x\right)=p{x}^2+qx+r$
$\alpha +\beta =-\frac{q}{p}$ and $\alpha \beta =\frac{r}{p}$, then the required polynomial has the zeroes as $\frac{1}{\alpha },\frac{1}{\beta }$
sum of its zeroes $=\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }$
$=\frac{\frac{-q}{p}}{\frac{r}{p}}=-\frac{q}{r}$
$=-\frac{b}{a}$ say
product of zeroes $=\frac{1}{\alpha },\frac{1}{\beta }=\frac{1}{\alpha }\beta =\frac{1}{\frac{r}{p}}=\frac{p}{r}=\frac{c}{a}$ say
if $a=1$, then $b=\frac{q}{r},c=\frac{p}{r}$ in $a{x}^2+bx+c$
the required polynomials is $(x^2+\frac{q}{r}x+\frac{p}{r})$