Question

if alpha and beta are the zeros of the quadratic polynomial f(x)=ax2+bx+c then evaluate 1/alpha3+1/beta3

Answer 1

Please see the attachment

Answer 2

Answer:

we are given with Quadratic polynomial , f(x) = ax² + bx + c and α & β are zeroes.

We use the relation of coefficient and zeroes.i.e.,

$\alpha+\beta=\frac{-b}{a}$

$\alpha\,\beta=\frac{c}{a}$

Consider,

$\frac{1}{\alpha^3}+\frac{1}{\beta^3}$

$\implies\frac{\alpha^3+\beta^3}{(\alpha\,\beta)^3}$

$\implies\frac{(\alpha+\beta)^3+3\apha\,\beta(\alpha+\beta)}{(\alpha\,\beta)^3}$

$\implies\frac{(\frac{-b}{a})^3+3\frac{c}{a}(\frac{-b}{a})}{(\frac{c}{a})^3}$

$\implies\frac{\frac{-b^3+3abc}{a^3}}{\frac{c^3}{a^3}}$

$\implies\frac{-b^3+3abc}{c^3}$

$\implies\frac{3abc-b^3}{c^3}$