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An alternative approach, which avoids the computation of at each step, is to compute by using integer arithmetic. Let

Now, has exactly two real conjugates corresponding to the two
conjugates and of . Since for all **k**,
it is easy to see that the conjugate of corresponding to
is . Let us denote this
conjugate by . For , has two real
conjugates and itself. By definition, . Since all other conjugates are nonreal, the sign of
the product of all conjugates is determined by that of , and thus

Since the determinant computed above is just the resultant of and , another alternative is to use the coefficient vector of to represent and replace the determinant computation by the computation of a resultant at each step. Or, combining the approaches of § 5.1 and § 5.2, one could compute using floating point except in ``delicate'' cases. Experiments showed that the approach of § 5.2 was generally considerably slower than that of § 5.1.

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