polynomial equation

Question

Find all real solutions of the polynomial equation. (Enter your answers as a comma-separated list.)

z⁴ + z³ + z² + 3z − 6 = 0

z=

Answer 1

Answer 2

z^4 +z^3 +z^2 +3z -6 =0

by descartes sign rule, 1 + real solution

and 1 or 3 - real solutions, 0 or 2 imaginary solutions

z=1 sort of jumps out at you, as the 1 positive real solution

then 1 or 3 negative real solutions

simplest method is use a graphing calculator, graph it and the x intercepts = the real solutions

= -2 and +1 = the 2 real solutions. the other 2 solutions are imaginary

divide (z+2)(z-1) into the 4D polynomial, then use the quadratic formula to find the conjugate pair of imaginary solutions: +/-isqr3