Question

Solve 3x^4-4x^3+7x^2+10x-4=0 in the complex number system.

Answer 1

Use the rational roots theorem to come up with the "potential" rational roots. So put the factors of the constant term over the leading coefficient.
Your possible rational roots are:
\[\frac{\pm 1,2,4}{1,3}\]
Then try them out, say you start with -1.
Do long division by (x+1)
If you get remainder ZERO then it is a root, and its whatever times the factor.
In this case, -1 is a root. The other root is (1/3)
So you will get a polynomial with two factors (x+1)(x-(1/3))(quadratic)
Then use the quadratic formula on the quadratic part.
Bam, all 4 roots. (The fundamental theorem of algebra guarantees n roots for an x^n polynomial)

Answer 2

http://www.wolframalpha.com/input/?i=solve++3x^4-4x^3%2B7x^2%2B10x-4%3D0