1. ## Factoring Complex Polynomials

Hi

I am asked to show all the seventh roots of -128 in polar form, no probs. But I am then asked

Factorise the polynomial $x^7+128$ into polynomial factors with
real coefficients, where the factors are either linear or quadratic. Give exact values for the coefficients, which may involve
trigonometric functions.

Not sure where to begin.

2. x^7 + 128 = x^7 + 2^7 = (x + 2)(x^6 - 2x^5 + 4x^4 - 8x^3 + 16x^2 - 32x + 64)

3. If you draw a picture you will see that there is one real root: -2 or and that the other roots come in pairs that are complex conjugates of each other. If you multiply the linear terms you get from each of these factors together you will get a quadratic polynomial with real coefficients. If you express the roots use a general trigonometric formula (no doubt involving various multiples of 2*pi/7 radians) it should be easy enough to get the required factorisation.

4. Real Root: x1 = -2

Complex Roots:

x2 = 2(-1)^(1/7),

x3 = -2(-1)^(2/7),

x4 = 2(-1)^(3/7),

x5 = -2(-1)^(4/7),

x6 = 2(-1)^(5/7),

x7 = -2(-1)^(6/7).

Now, you have an idea how to convert that to polar form as suggested by the the graph