Aus Online Mathematik Brückenkurs 2

Because \displaystyle z=1-2i should be a root of the equation, we can substitute \displaystyle z=1-2i in and the equation should be satisfied,

\displaystyle (1-2i)^3 + a(1-2i) + b = 0\,\textrm{.}

We will therefore adjust the constants \displaystyle a and \displaystyle b so that the relation above holds. We simplify the left-hand side,

\displaystyle -11+2i+a(1-2i)+b=0

and collect together the real and imaginary parts,

\displaystyle (-11+a+b)+(2-2a)i=0\,\textrm{.}

If the left-hand side is to equal the right-hand side, the left-hand side's real and imaginary parts must be equal to zero, i.e.

\displaystyle \left\{\begin{align} -11+a+b &= 0\,,\\[5pt] 2-2a &= 0\,\textrm{.} \end{align}\right.

This gives \displaystyle a=1 and \displaystyle b=10.

The equation is thus

\displaystyle z^3+z+10=0

and has the prescribed root \displaystyle z=1-2i.

What we have is a polynomial with real coefficients and we therefore know that the equation has, in addition, the complex conjugate root \displaystyle z=1+2i.

Hence, we know two of the equation's three roots and we can obtain the third root with help of the factor theorem. According to the factor theorem, the equation's left-hand side contains the factor

\displaystyle \bigl(z-(1-2i)\bigr)\bigl(z-(1+2i)\bigr)=z^2-2z+5

and this means that we can write

\displaystyle z^3+z+10 = (z-A)(z^2-2z+5)

where \displaystyle z-A is the factor which corresponds to the third root \displaystyle z=A. Using polynomial division, we obtain the factor

\displaystyle \begin{align} z-A &= \frac{z^3+z+10}{z^2-2z+5}\\[5pt] &= \frac{z^3-2z^2+5z+2z^2-5z+z+10}{z^2-2z+5}\\[5pt] &= \frac{z(z^2-2z+5)+2z^2-4z+10}{z^2-2z+5}\\[5pt] &= z + \frac{2z^2-4z+10}{z^2-2z+5}\\[5pt] &= z + \frac{2(z^2-2z+5)}{z^2-2z+5}\\[5pt] &= z+2\,\textrm{.} \end{align}

Thus, the remaining root is \displaystyle z=-2.