Question

(a) Find the real root of the equation z3 + z + 10 = 0 given that one root is 1 − 2i.
(b) Given that 3 + i is a root of the equation z3 − 3z2 − 8z + 30 = 0, find the
remaining roots.

Answer 1

a)10 have next dividers: $\pm 1;\pm2; \pm 5; \pm10;$

$For \space 1:\newline (1)^3+1+10\not=0\newline For \space -1:\newline (-1)^3+(-1)+10\not=0\newline For \space 2:\newline (2)^3+2+10\not=0\newline For \space -2:\newline (-2)^3+(-2)+10=0\newline$

So z=-2 is a root of this equation

b)$\dfrac{z^3-3z^2-8z+30}{z-3-i}=z^2+iz+(-9+3i)\newline$

$z^2+iz+(-9+3i)=0\newline D=i^2-4(-9+3i)=35-12i\newline \sqrt{D}=\sqrt{35-12i}\newline z_1=\dfrac{-i+\sqrt{35-12i}}{2}\newline z_2=\dfrac{-i-\sqrt{35-12i}}{2}\newline$