Find the roots of the polynomial $ 3X^3 -32X^2+73X +28$ using Vieta's relations

Question

They are asking me to find the roots of the polynomial $ 3X^3 -32X^2+73X +28$ using Vieta's relations. They also tell me that $x_1 - x_2 = 3$.

I have tried to use first Vieta's relation($x_1 + x_2 + x_3 = \frac{32}{3}$), but I don't get anything meaningful.

Answer 1

Using your first relation, $x_1+x_2+x_3=\frac{32}{3}$, and given that $x_1-x_2=3$, so that $x_1=3+x_2$ we can obtain $$x_3=\frac{23}{3}-2x_2$$ A second Vieta's relation can then be used $$x_1x_2+x_1x_3+x_2x_3=\frac{73}{3}$$ Substituting $x_1=3+x_2$ and $x_3=\frac{32}{3}-2x_2$ into this results in a quadratic in $x_2$, which can be simplified to $$(x_2-4)(9x_2-1)=0$$ resulting in two possible values, $x_2=4$ or $x_2=\frac{1}{9}$. Substituting these two values into the cubic leads to only one of them, $x_2=4$, satisfying the equation.

Thus, we have $x_1=7$ and $x_3=-\frac{1}{3}$ as the other roots.

Summarising, the roots of the polynomial are $x_1=7$, $x_2=4$ and $x_3=-\frac{1}{3}$.