# How do you find all the real and complex roots of #x^4 - 2x^3 - 4x^2 - 8x - 32 = 0#?

##### 1 Answer

Use the rational root theorem, then divide by the factors found to find the remaining roots.

#x=-2# ,#x=4# ,#x=+-2i#

#### Explanation:

Let

By the rational root theorem, the only possible rational roots of

That means that the only possible rational roots are factors of

#+-1# ,#+-2# ,#+-4# ,#+-8# ,#+-16# ,#+-32#

Trying each of these in turn, we find:

#f(-2) = 16+16-16+16-32 = 0#

#f(4) = 256-128-64-32-32 = 0#

So

#x^4-2x^3-4x^2-8x-32#

#= (x+2)(x^3-4x^2+4x-16)#

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

#= (x+2)(x-4)(x-2i)(x+2i)#