1)
Solve for x:
x^3-1 = 0
Add 1 to both sides:
x^3 = 1
Taking cube roots gives 1 times the third roots of unity:
Answer: | x = 1 or x = -(-1)^(1/3) or x = (-1)^(2/3)
2)
Solve for x:
x^4-1 = 0
Add 1 to both sides:
x^4 = 1
Taking 4^th roots gives 1 times the 4^th roots of unity:
Answer: | x = -1 or x = -i or x = i or x = 1
3)
Solve for x:
x^3-2 x^2-5 x+6 = 0
The left hand side factors into a product with three terms:
(x-3) (x-1) (x+2) = 0
Split into three equations:
x-3 = 0 or x-1 = 0 or x+2 = 0
Add 3 to both sides:
x = 3 or x-1 = 0 or x+2 = 0
Add 1 to both sides:
x = 3 or x = 1 or x+2 = 0
Subtract 2 from both sides:
Answer: | x = 3 or x = 1 or x = -2
4)
Solve for x:
x^4+2 x^3-16 x^2-2 x+15 = 0
The left hand side factors into a product with four terms:
(x-3) (x-1) (x+1) (x+5) = 0
Split into four equations:
x-3 = 0 or x-1 = 0 or x+1 = 0 or x+5 = 0
Add 3 to both sides:
x = 3 or x-1 = 0 or x+1 = 0 or x+5 = 0
Add 1 to both sides:
x = 3 or x = 1 or x+1 = 0 or x+5 = 0
Subtract 1 from both sides:
x = 3 or x = 1 or x = -1 or x+5 = 0
Subtract 5 from both sides:
Answer: | x = 3 or x = 1 or x = -1 or x = -5