Question

If alpha and beta are zeroes of polynomial 3x2 5x 13 then evaluate alpha square upon beta plus beta square upon alpha

Answer 1

Given:

α and β are zeros of polynomial $3^{2} + 5x + 13$

To Find:

$\frac{\alpha ^{2} }{\beta } + \frac{\beta ^{2} }{\alpha }$ = ?

Solution:

For a quadratic equation ,

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

Product of Zeros = $\alpha \beta = \frac{ c}{a}$

For the given quadratic equation a = 3 b = 5 c = 13

$\alpha \beta = \frac{c}{a} = \frac{13}{3}$

$\alpha + \beta = \frac{-b}{a} = \frac{-5}{3}$

Evaluate  $\frac{\alpha ^{2} }{\beta } + \frac{\beta ^{2} }{\alpha }$

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

$(\alpha + \beta )^{3} = \alpha ^{3} + \beta ^{3} + 3\alpha \beta (\alpha + \beta )\\\alpha ^{3} + \beta ^{3} = (\alpha + \beta )^{3} - 3\alpha \beta (\alpha + \beta )\\\frac{\alpha ^{3} + \beta ^{3} }{\alpha \beta } = \frac{(\alpha + \beta )^{3} - 3\alpha \beta (\alpha + \beta )\\ }{\alpha\beta }$

Substitute the values of

$\alpha + \beta = \frac{-5}{3} \\\alpha \beta = \frac{13}{3}$

$\frac{\alpha ^{3} + \beta ^{3} }{\alpha\beta } = \frac{460}{117}$

The value of $\frac{\alpha ^{3} + \beta ^{3} }{\alpha\beta }$ is equal to $\frac{460}{117}$ .