Exercise 3.

Find all rational zeros of the polynomial

displaymath60

Answer.

Rational zeros of the form tex2html_wrap_inline64 satisfy the following: p divides the constant coefficient 4; q divides the leading coefficient 3. The choices for p are: tex2html_wrap_inline76 , the only choices for q are: tex2html_wrap_inline80 . Thus p/q is one of the following:

displaymath62

Check each of these 12 candidates by plugging the values into the polynomial. You see that only tex2html_wrap_inline84 evaluates to 0.

If you have a graphing utility, you can reduce the checking:

Here is the graph again after zooming in on the root:

The only candidate left is tex2html_wrap_inline84 . Since tex2html_wrap_inline90 , x=-2/3 is indeed a rational zero.

The only rational zero is tex2html_wrap_inline84 .

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Helmut Knaust
Tue Jun 24 12:43:57 MDT 1997

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