Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function P(x)=18x3+3x2−4x−1. Use Descartes' Rule of signs, the Upper and Lower Bounds Theorem, the Quadratic Formula or other factoring techniques.
The possible rational zeros of P are the factors of 1 divided by the factors of 18 which are ±1,±12,±13,±16,±19 and ±118. Then, by applying Synthetic Division and trial and error,
Again, by applying synthetic division
Thus,
P(x)=18x3+3x2−4x−1=(x+13)(18x2−3x−3)=(x+13)(x+13)(18x−9)=(x+13)2(18x−9)
Thus, the zeros of P are −13 and 12. The zeros have multiplicity of 2 and 1 respectively.
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