Question

If alpha and Beta are the zeroes of the quadratic polynomial ax²+bx +c,then find a quadratic polynomial whose zeroes are 1/alpha and 1/beta

Answer 1

Dear student
$$\begin{gathered} \sin ce\alpha and\beta arethezerosofthepolynomialf\left(x\right)=a{x}^2+bx+c \\ \therefore \alpha +\beta =-\frac{b}{a} \\ and\alpha \beta =\frac{c}{a} \\ LetSandPdenoterespectivelythesumandtheproductofzerosoftherequiredpolynomial. \\ Then, \\ S=\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\beta +\alpha }{\alpha \beta }=\frac{{\displaystyle \frac{-b}{a}}}{{\displaystyle \frac{c}{a}}}=\frac{-b}{c} \\ P=\frac{1}{\alpha }\times \frac{1}{\beta }=\frac{1}{\alpha \beta }=\frac{a}{c} \\ Hencerequiredpolynomialis \\ g\left(x\right)=k\left\{x^2-Sx+P\right\}=k\left\{x^2+\frac{b}{c}x+\frac{a}{c}\right\}wherekisanynon-zerocons\tan t. \end{gathered}$$
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