Question

Consider the following complex numbers:
1 + i
1.5 + i
1.5 − i
1 + 0.5 i
−1 + i
1 − 0.5 i
How many are a fourth root of −4?

Answer 1

i know 1 + i is

Answer 2

any others??

Answer 3

Finding the fourth root of -4 is the same as solving the equation

x^4 = -4


which becomes


x^4 + 4 =0


So it's a polynomial of degree 4. So there will be 4 complex roots. Luckily, complex roots come in conjugate pairs. So if we know that 1+i is a root (from cwrw238), then 1-i is also a root.


This means that


x = 1+i or x = 1-i


x - (1+i) = 0 or x-(1-i) = 0


(x - (1+i))(x-(1-i)) = 0


( (x-1) - i)( (x-1) +i) = 0


(x-1)^2 - i^2 = 0


(x-1)^2 + 1 = 0


x^2 - 2x+1+1 = 0


x^2 - 2x + 2=0


So this means that x^2 - 2x + 2 is a factor of x^4 + 4


To find the other factor, use polynomial long division

x^2 + 2x + 2
------------------------------
x^2 - 2x + 2 | x^4 + 0x^3 + 0x^2 + 0x + 4
x^4
- 2x^3 + 2x^2
----------------
2x^3 - 2x^2
2x^3 - 4x^2 + 4x + 4
-----------------
2x^2 - 4x + 4
2x^2 - 4x + 4
--------------
0

This does two things really:
First is the remainder of 0 confirms that x^2-2x+2 is really a factor of x^4 + 4. Second is that the quotient x^2+2x+2 shows us that


x^4 + 4 = (x^2-2x+2)(x^2+2x+2)


From here, all you need to do is solve x^2+2x+2 = 0 to find the other two roots.


Use the quadratic formula to solve


x = (-b+-sqrt(b^2-4ac))/(2a)


x = (-(2)+-sqrt((2)^2-4(1)(2)))/(2(1))


x = (-2+-sqrt(4-(8)))/(2)


x = (-2+-sqrt(-4))/2


x = (-2+sqrt(-4))/2 or x = (-2-sqrt(-4))/2


x = (-2+2*i)/2 or x = (-2-2*i)/2


x = -1 + i or x = -1-i



So in total, there are 4 roots and they are


1+i, 1-i, -1+i, -1-i