Find a quadratic polynomial whose zeros are reciprocals of the zero of the polynomial `f(x)=a x^2+b x+c , a!=0 ,c!=0.`

Explanation

Let `alpha and beta` be zero of the polynomial `f(x)=ax^2+bx+c, a≠0, c≠0`.

`therefore, alpha + beta = (-b)/a`

Also, `alpha beta = c/a`

`therefore 1/alpha + 1/beta = (alpha + beta)/(alpha beta) = ((-b)/a)/(c/a) = -b/c`

And, `1/alpha * 1/beta = 1/(alpha beta) = a/c`

A quadratic polynomial whose zeros are `1/alpha and 1/beta` is `x^2 – (1/alpha + 1/beta)x + 1/alpha * 1/beta`.

Thus, polynomial is `x^2 + (b/c)x + a/c`.