I was given this equation. $x^3 + 3px^2 + qx + r=0$. The roots are $1, -1$, and $3$.
Ive tried dividing the equation by $(x-1)$ to get a quadratic to make it easier for me. But that ended up really badly.
I also inputted the different roots into the equation to get different equations that I could solve. When I tried to prove my answer it turned out to be a flop.
And I also tried multiplying the three factors and comparing coefficients. It didnt seem right.
Best Answer
By Vieta's relations between the roots $\,a=1,b=-1,c=3\,$ and coefficients $\,1,3p,q,r\,$:
$\;\;-3p=a+b+c=1+(-1)+3=3$
$\;\;q = ab+bc+ca=1 \cdot (-1) + (-1) \cdot 3 + 3 \cdot 1 = -1$
$\;\;-r = abc = 1 \cdot (-1) \cdot 3 = -3$