Calculating complex roots "Find all the complex roots of the following polynomials
A) $S(x)=135x^4 -324x^3 +234x^2 -68x+7$, knowing that all its real roots belong to the interval $(0.25;1.75)$
B)$M(x)=(x^3 -1+i)(5x^3 +27x^2 -28x+6)$
"
Well, in A) I don't know how to use the given information about real roots. I mean, I know that I can apply Bolzano but I don't think that's very useful. To find the complex roots I should have some information about a complex root in particular so that I could use Ruffini, but this is not the case.
And in B) I know that $(x^3 -1+i)$ is giving me some information related to a complex root, but that "^3" bothers me. If it wasn't there, I would know that $1-i$ is a root...

Since $\frac{1}{3}$ is a root of $S$, we obtain:
$$S=135x^4-324x^3+234x^2-68x+7=$$
$$=135x^4-45x^3-279x^3+93x^2+141x^2-47x-21x+7=$$
$$=(3x-1)(45x^3-93x^2+47x-7)=$$
$$=(3x-1)(45x^3-15x^2-78x^2+26x+21x-7)=$$
$$=(3x-1)^2(15x^2-26x+7)=(3x-1)^2(15x^2-5x-21x+7)=(3x-1)^3(5x-7).$$
Since $\frac{3}{5}$ is a root of $5x^3+27x^2-28x+6$, we obtain:
$$5x^3+27x^2-28x+6=5x^3-3x^2+30x^2-18x-10x+6=(5x-3)(x^2+6x-2)=$$
$$=(5x-3)((x+3)^2-11)=(5x-3)(x+3-\sqrt{11})(x+3+\sqrt{11}).$$
Also, $$\sqrt[3]{1-i}=\sqrt[6]2\sqrt[3]{\cos315^{\circ}+i\sin315^{\circ}}=$$
$$=\sqrt[6]2(\cos(105^{\circ}+120^{\circ}k)+i\sin(105^{\circ}+120^{\circ}k)),$$
where $k\in\{0,1,2\}$.
