**Unformatted text preview: **+ a1 x + a0
is a polynomial of degree n with n ≥ 1, and a0 , a1 , . . . an are integers. If r is a rational zero of
f , then r is of the form ± p , where p is a factor of the constant term a0 , and q is a factor of the
q
leading coeﬃcient an .
The rational zeros theorem gives us a list of numbers to try in our synthetic division and that
is a lot nicer than simply guessing. If none of the numbers in the list are zeros, then either the
polynomial has no real zeros at all, or all of the real zeros are irrational numbers. To see why the
Rational Zeros Theorem works, suppose c is a zero of f and c = p in lowest terms. This means p
q
and q have no common factors. Since f (c) = 0, we have
an p
q n + an−1 p
q n−1 + . . . + a1 p
q + a0 = 0. 1
Carl is the purist and is responsible for all of the theorems in this section. Jeﬀ, on the other hand, has spent too
much time in school politics and has been polluted with notions of ‘compromise.’ You can blame the slow decline of
civilization on him and those like him who mingle mathematic...

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