SOLUTION: Factor the polynomial function over the complex numbers. {{{ f(x) = x^4 - x^3 - 2x - 4 }}} f(x) =

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Factor the polynomial function over the complex numbers. {{{ f(x) = x^4 - x^3 - 2x - 4 }}} f(x) =      Log On


   

Question 1127861: Factor the polynomial function over the complex numbers.
+f%28x%29+=++x%5E4+-+x%5E3+-+2x+-+4+
f(x) =
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Remember that "complex numbers" includes real numbers. So look for rational roots first.

Substitution show f(-1)=0, so -1 is a root. Extract that root using synthetic division.

   -1 |  1 -1  0 -2 -4
      |    -1  2 -2  4
      ----------------
        1  -2  2 -4  0

The remaining polynomial is x%5E3-2x%5E2%2B2x-4. A second real root can be found using factoring by grouping.



The remaining quadratic factor can be factored over the complex numbers as

%28x%2Bi%2Asqrt%282%29%29%28x-i%2Asqrt%282%29%29

So the complete factorization over the complex numbers is

x%5E4-x%5E3-2x-4+=+%28x%2B1%29%28x-2%29%28x%2Bi%2Asqrt%282%29%29%28x-i%2Asqrt%282%29%29