Question

If alpha,beta are zeros of polynomial f(x)= ax^2 +bx + c, then evaluate 1/alpha^3+ 1/beta^3​

Answer 1

GIVEN

α and β are the zeros of the polynomial f(x)

$f(x)=ax^{2} +bx+c$

TO FIND :- $\frac{1}{\alpha ^{3} } +\frac{1}{\beta ^{3} }$

FORMULA

α+β= -b/a

αβ = c/a

SOLUTION

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

$=\frac{\alpha^{3}+\beta ^{3}}{\alpha \beta }\\\\=\frac{({\alpha+ \beta) }^{3}- 3\alpha \beta(\alpha+ \beta)}{\alpha \beta}\\\\=\frac{\frac{-b}{a}-3(\frac{c}{a})(\frac{-b}{a})}{\frac{c}{a}} \\\\\\=\frac{\frac{-b}{a}+\frac{3cb}{a^{2 }}}{\frac{c}{a} } \\\\=\frac{\frac{-ba+3cb}{a^{2}}}{\frac{c}{a} }\\\\=\frac{-ba+3bc}{ca} [ANS]$