Question

Knowing that 2 and 3 are the roots of the equation $2x^3+m{x}^2-13x+n=0,$ determine m+n+2(p)?where p is the third root of the given equation.

Answer 1

As $2$ and $3$ are the roots of the given equation we have $4m+n=10,$ and $9m+n=-15.$
Solving, we get $m=-5$
$\therefore n=30.$
Hence the given equation is
$2x^3-5x^2-13x+30=0$
Now $\alpha +\beta +\gamma =5/2\Rightarrow 2+3+\gamma =5/2$
$\therefore \gamma =5/2-5=-5/2$
$\therefore m+n+2\left(p\right)=-5+30+2(\frac{-5}{2})=-5+30-5=20$