Factoring Polynomial z^4-4z^3+6z^2-4z-15 =0

[Fabio010]

z^4-4z^3+6z^2-4z-15 =0

How can i factor this polynomial in order to find the solutions??


I tried with the ruffini' rule.

and i reached the following equation [(z+1)(-z^3-5z^2+11z-15)] =0

now how can i factor (-z^3-5z^2+11z-15) ?

i tried it, but i can not solve it... :/

 

[alberto7]

Proceed with Ruffini. You'll find another root (because the problem is easy) and the remaining factor is quadratic, whose solutions you get with the formula.

 

[Fabio010]

the solutions are

-1; 3; 1+/-2i

i am going to try with ruffini again.

 

[Fabio010]

i cant. Even knowing the solutions, i can not proceed with ruffini's rule.
Maybe something is escaping me.

 

[alberto7]

Redo the quotient (z^4-4z^3+6z^2-4z-15)/(z+1), since the leading term must be z^3, not -z^3.

 

[Fabio010]

ok it now makes sense.

now i factor it


(z^3-5z^2+11z-15)/(z-3)



...

but without the solution i would never be able to discover that i should divide (z^3-5z^2+11z-15) by (z-3)

 

[alberto7]

Do Ruffini again: try with the divisors of -15 of both signs.