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Aus Online Mathematik Brückenkurs 2

If the equation has the root z=1, this means, according to the factor rule, that the equation mustcontain the z=1, i.e. the polynomial on the left-hand side can be written as

z3−3z2+4z−2=z2+Az+Bz−1 

for some constants A and B. We can determine the other unknown factor using polynomial division:

z3−3z2+4z−2=z2+Az+Bz−1z2+Az+B=z−1z3−3z2+4z−2=z−1z3−z2+z2−3z2+4z−2=z−1z2z−1−2z2+4z−2=z2+z−1−2z2+4z−2=z2+z−1−2z2+2z−2z+4z−2=z2+z−1−2zz−1+2z−2=z2−2z+z−12z−2=z2−2z+z−12z−1=z2−2z+2

Thus, the equation can be written as

z−1z2−2z+2=0 

The advantage of writing the equation in this factorized form is that we can now conclude that the equation's two other roots must be zeros of the factor z2−2z+2. This is because the left-hand side is zero only when at least one of the factors z−1 or z2−2z+2 is zero, and we see directly that z−1 is zero only when z=1.

Hence, we determine the roots by solving the equation

z2−2z+2=0

Completing the square gives

z−12−12+2=0z−12=−1

and taking the root gives that z−1=i i.e. z=1−i and z=1+i.

The equation's other roots are z=1−i and z=1+i.

As an extra check, we investigate whether z−1=i really are roots of the equation.

z=1+i:z3−3z2+4z−2=z−3z+4z−2=1+i−31+i+41+i−2=−2+i1+i+41+i−2=−2+i−2i−1+41+i−2=1−i1+i−2=12−i2−2=1+1−2=0

z=1−i:z3−3z2+4z−2=z−3z+4z−2=1−i−31−i+41−i−2=−2−i1−i+41−i−2=−2−i+2i−1+41−i−2=1+i1−i−2=12−i2−2=1+1−2=0

NOTE: Writing

z3−3z2+4z−2=z−3z+4z−2 

is known as the Horner scheme and is used to reduce the amount of the arithmetical work.