Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function P(x)=x4+7x3+9x2−17x−20. Use Descartes' Rule of signs, the Upper and Lower Bounds Theorem, the Quadratic Formula or other factoring techniques.
The possible rational zeros of the polynomial P are the factors of 20 which are ±1,±2,±4,±5,±10,±20. Then, by using Synthetic Division and trial and error
Again, by applying Synthetic Division
Thus,
P(x)=x4+7x3+9x2−17x−20=(x+4)(x3+3x2−3x−5)=(x+4)(x+1)(x2+2x−5)
To get the remaining zeros, we use quadratic formula.
x=−b±√b2−4ac2a=−(2)±√22−4(1)(−5)2(1)=−1±√6
Therefore, the zeros of P are −4,−1,−1+√6 and −1−√6. Each zeros have multiplicity of 1.
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