Hello

The question is Factorise z^3 +i over C . (Book's answer is (z-i)(z^2+iz-1))

I have attempted it 3 ways: Number 1 seems to work, but cannot get 2 or 3 to work.

Sorry I didnt know how to do long division in word so did it by hand instead.

In (1) I tried to divide z^3+i by (z-i) which I think is a factor

and this happened to give the same answer as the book.

**You are correct, but you should have written above that P(i)=0**
In (2) I tried to divide z^3+i by (z+i) which I think is also a factor but I couldnt not complete the division properly.

**Your division is correct and if it was a factor, your remainder would be 0.**

The remainder is not zero so that is not a factor of the complex polynomial.
In (3) I attempted to use this method :

but I do know now how to this method with this question.

**Notice this method states that the coefficients must be real (see "Think 1").**
Could you please correct my working and show me how to use method 3 if possible as I think that is the method the teacher wants us to do for these types of questions.

Thanks

*note: should have read + i not +1 in title.