SOLUTION: with the help of the quadratic formula, find all the complex roots of each of the polynomials given in exercise 1. iii) z^3 + i

Question 194674This question is from textbook An Introduction to Abstract Algebra with Notes to the Future Teacher
: with the help of the quadratic formula, find all the complex roots of each of the polynomials given in exercise 1.
iii) z^3 + i This question is from textbook An Introduction to Abstract Algebra with Notes to the Future Teacher

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
First, perform synthetic division on the polynomial using the test zero . Note: you can easily solve for "z" and find that one solution is

i |  1  0  0   i
  |     i  -1  -i
  -----------
     1  i  -1   0

Since the first three numbers in the third row are 1, i, and -1, this means that

Now simply use the quadratic formula to solve for "z":

Start with the quadratic formula

Plug in , , and

Square to get .

Multiply

Combine like terms.

So the other two solutions are or

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Answer:

So the 3 solutions are , , or

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