Question

Find a quadratic polynomial whose zeroes are reciprocals of the zeroes of the polynomial f(x) = ax2 + bx + c, a ≠ 0, c ≠ 0.

Comments

Answer please

Answer 1

Let p,q be zeros of ax²+bx+c

Answer 2

(without using the vieta's formulas)

let \(p,q\) be roots of \(f(x)\).    (\(x=p\) , \(x=q\))

let  \(X=\dfrac{1}{x}\) , ( when \(x=p\) , \(X=\dfrac{1}{p}\)and \(x=q\) , \(X=\dfrac{1}{q}\) )

if we can find a quadratic equation of \(X\) we are done. (then its roots are \(\dfrac{1}{p}\) , \(\dfrac{1}{q}\))

\(X=\dfrac{1}{x}\)  ⇒  \(x=\dfrac{1}{X}\)

\(ax^2+bx+c=0\)

\(a(\dfrac{1}{X})^2+b(\dfrac{1}{X})+c=0\)

⇒ \(a+bX+cX^2=0\)  (multiply by \(X^2\))

hence the required quadratic equation is \(cX^2+bX+a=0\)