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Lösung 3.4:5

Aus Online Mathematik Brückenkurs 2

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A polynomial is said to have a triple root z=c if the equation contains the factor zc3 


For our equation, this means that the left-hand side can be factorized as


z46z2+az+b=zc3zd 


according to the factor theorem, where z=c is the triple root and z=d is the equation's fourth root (according to the fundamental theorem of algebra, a fourth-order equation always has four roots, taking into account multiplicity).

We will now try to determine a, b, c and d so that both sides in the factorization above agree.

If we expand the right-hand side above, we get


zc3zd=zc2zczd=z22cz+c2zczd=z33cz2+3c2zc3zd=z43c+dz3+3cc+dz2c2c3dz+c3d


and this means that we must have


z46z2+az+b=z43c+dz3+3cc+dz2c2c3dz+c3d 


Because two polynomials are equal if an only if their coefficients are equal, this gives


3c+d=03cc+d=6c2c3d=ac3d=b


From the first equation, we obtain d=3c and substituting this into the second equation gives us an equation for c,


3cc3c=66c2=6 


i.e. c=1 or c=1. The relation d=3c gives that the corresponding values for d are d=3 and d=3. The two last equations give us the corresponding values for a and b,


c=1 d=3:a=12133=8b=133=3 


c=1 d=3:a=12133=10b=133=3


Therefore, there are two different answers:


a=8 and b=3 give the triple root z=1 and the single root z=3;

a=10 and b=3 give the triple root z=1 and the single root z=3.