Exercise 2.

Find all rational zeros of the polynomial

displaymath30

Answer.

Rational zeros of the form tex2html_wrap_inline34 satisfy the following: p divides the constant coefficient 4; q divides the leading coefficient 1. The choices for p are: tex2html_wrap_inline46 , the only choices for q are: tex2html_wrap_inline50 . Thus p/q is one of the following:

displaymath32

Check each of these 6 candidates by plugging the values into the polynomial: P(1)=-1, P(-1)=-21, P(2)=-12, P(-2)=-28, P(4)=44, P(-4)=204. None of these candidates satisfies P(x)=0; consequently this polynomial does not have any rational roots. All the work was in vain.

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Helmut Knaust
Tue Jun 24 12:41:53 MDT 1997

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