Question

Find a quadratic polynomial whose zeroes are reciprocals of the zero of the polynomial f(x) = ax^2 + bx + c, a ≠ 0, c≠ 0.

Answer 1

cx^2+bx+a=0

Answer 2

Let the zeros of $f(x)=ax^2+bx+c$ be $r$ and $s$. Then:

$$r+s=-\frac{b}{a},\quad rs=\frac{c}{a}.$$

The reciprocals $1/r$ and $1/s$ have:

$$\text{Sum}=\frac{1}{r}+\frac{1}{s}=\frac{r+s}{rs}=\frac{-\frac{b}{a}}{\frac{c}{a}}=-\frac{b}{c},\quad \text{Product}=\frac{1}{rs}=\frac{a}{c}.$$

A quadratic polynomial with these as zeros is:

$$x^2 - \left(-\frac{b}{c}\right)x + \frac{a}{c}=x^2+\frac{b}{c}x+\frac{a}{c}=0.$$

Multiplying through by $c$ (since $c\ne0$) gives:

$$cx^2+bx+a=0.$$