Monday, December 5, 2011

College Algebra, Chapter 4, Chapter Review, Section Review, Problem 54

Find all rational, irrational and complex zeros (and state their multiplicities) of the polynomial function P(x)=x4+7x3+9x217x20. 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+9x217x20=(x+4)(x3+3x23x5)=(x+4)(x+1)(x2+2x5)


To get the remaining zeros, we use quadratic formula.


x=b±b24ac2a=(2)±224(1)(5)2(1)=1±6


Therefore, the zeros of P are 4,1,1+6 and 16. Each zeros have multiplicity of 1.

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