Question

Find the quadratic polynomial whose zeros are reciprocals of the zeros of the polynomials $f\left(x\right)=a{x}^2+bx+c,a\ne 0,c\ne 0$.

Answer 1

Let p,q be zeros of $a{x}^2+bx+c$

$\therefore p+q=\frac{-b}{a}$ & $pq=\frac{c}{a}$

Let P & Q be zeros of required polynomial

It is given that P= 1/p , Q=1/q

Then,

$P+Q=\frac{1}{p}+\frac{1}{q}=\frac{q+p}{pq}=\frac{-b/a}{c/a}=\frac{-b}{c}$

$PQ=\frac{1}{p}.\frac{1}{q}=\frac{1}{pq}=\frac{a}{c}$

$\therefore$ Required polynomial = $x^2+\frac{b}{c}x+\frac{a}{c}\Rightarrow c{x}^2+bx+a$