Find all complex zeros

Question

Find all the complex zeros of f(x)=5x^4-4x^3+19x^2-16x-4. Give exact values

Answer 1

Using p/q to find possible roots (factors of -4 and factors of 5), first try +1 (1/1).

Using synthetic division:

-1 | 5    -4   19  -16 -4

 

             5     1   20   4

 

      5     1    20    4   0


So 1 is a root. It's not complex, however.


Using the bottom row of the synthetic division problem above as the coefficients for the other factor besides (x-1):

5x³ + 1x² + 20x +4 would therefore be another factor.


Factor 5x³ + 1x² + 20x +4 :

x²(5x+1) + 4(5x+1)

(x² + 4)(5x+1) are factors.


Since 5x + 1 = 0 gives you -1/5 as a root,

only x² + 4 = 0 can produce complex roots, if any.


x² + 4 = 0

x² = -4

x = +/- sqrt(-4)

x = {+2i, -2i}