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

Question

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

Accepted Answer

Step 1: Let zeroes of $f(x)$ be $\alpha$ and $\beta$.
$\alpha + \beta = -\frac{b}{a}$, $\alpha\beta = \frac{c}{a}$
Step 2: New sum of zeroes $= \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = -\frac{b}{c}$
Step 3: New product of zeroes $= \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha\beta} = \frac{a}{c}$
Step 4: Required polynomial $= x^2 + \frac{b}{c}x + \frac{a}{c}$ or $(cx^2 + bx + a)$

Question

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