factoring over complex numbers

Question

	Factor  over the complex numbers.	a.
b.
c.
d.

Answer 1

When you see an expression like x³ + k, a likely factor is x ± one of the factors of k.

Suppose you had x3 + 1. to see if x + 1 is a factor, you can do the long division.

Here we have an expression using i, which is the square root of -1. The process is the same, but just remember the special rules for i:

i² = -1

i³ = -i

i⁴ = +1

etc.

The long division for x³ - i / x - i:

1st term: x²

multiply by x - i to get x³ - ix²

subtract to get ix²

2nd term: ix

multiply by x - i: ix² + x

subtract to get -x

bring down the -1 to get -x - 1

x - i does not go into this without a remainder, so this is not the right solution

Answer 2

Recall: A³ + B³ = (A + B)(A² - AB + B²)

-i = i³

So, x³ - i = x³ + i³ = (x + i)(x² -xi + i²) = (x + i)(x² - xi - 1)