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Unformatted text preview: we say that
it is factored completely over the complex numbers, meaning that it is impossible to factor
the polynomial any further using complex numbers. If we wanted to completely factor f (x) over
the real numbers then we would have stopped short of ﬁnding the nonreal zeros of f and factored
f using our work from the synthetic division to write f (x) = x − 2
x + 3 12x2 − 12x + 12 ,
or f (x) = 12 x − 1
x + 1 x2 − x + 1 . Since the zeros of x2 − x + 1 are nonreal, we call
2 − x + 1 an irreducible quadratic meaning it is impossible to break it down any further using
real numbers. The last two results of the section show us that, at least in theory, if we have a
polynomial function with real coeﬃcients, we can always factor it down enough so that any nonreal
zeros come from irreducible quadratics.
Theorem 3.15. Conjugate Pairs Theorem: If f is a polynomial function with real number
coeﬃcients and z is a zero of f , then so is z .
To prove the theorem, suppose f is a polynomial with real number coeﬃcients. Speciﬁcally, let
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