Lösung 3.4:5
Aus Online Mathematik Brückenkurs 2
A polynomial is said to have a triple root
z−c
3
For our equation, this means that the left-hand side can be factorized as
z−c
3
z−d
according to the factor theorem, where
We will now try to determine
If we expand the right-hand side above, we get
z−c
3
z−d
=
z−c
2
z−c
z−d
=
z2−2cz+c2
z−c
z−d
=
z3−3cz2+3c2z−c3
z−d
=z4−
3c+d
z3+3c
c+d
z2−c2
c−3d
z+c3d
and this means that we must have
3c+d
z3+3c
c+d
z2−c2
c−3d
z+c3d
Because two polynomials are equal if an only if their coefficients are equal, this gives
3c+d=03c
c+d
=−6−c2
c−3d
=ac3d=b
From the first equation, we obtain
c−3c
=−6−6c2=−6
i.e.
d=−3:a=−12
1−3
−3
=8b=13
−3
=−3
d=3:a=−
−1
2
−1−3
3
=10b=
−1
3
3=−3
Therefore, there are two different answers:
a=8
a=10