If α and β are the roots of the equation $3x^2+5x+4=0$, find the value of $α^3+β^3$ If α and β are the roots of the equation $3x^2+5x+4=0$, find the value of $α^3+β^3$
How can I factorize the expression to use the rule of sum and product of roots?
The answer is $\frac{55}{27}$

Rename $a=\alpha$ and $b=\beta$
Since $a$ is a solution of $3x^2+5x+4=0$ we have $3a^2+5a+4=0$ and thus $a^2={1\over 3}(-5a-4)$, so:
$$ 3a^3 = -5a^2-4a = -{5\over 3}(-5a-4) -4a = {13\over 3}a+{20\over 3}$$
the same is true for $b$, so we have
$$9a^3+9b^3 = 13(a+b)+40 = 13{-5\over 3 }+40 = {55\over 3}$$
and thus conclusion.   
