Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(t\right)=t^2-4t+3$, find the value of ${\alpha }^4{\beta }^3+{\alpha }^3{\beta }^4$.

Answer 1

$f\left(t\right)=t^2-4t+3$

Roots are $\alpha$ and $\beta$

$\alpha +\beta =4$

$\alpha \beta =3$

${\alpha }^4{\beta }^3+{\alpha }^3{\beta }^4={\alpha }^3{\beta }^3(\alpha +\beta )$

$3^3\left(4\right)$

$=108$