Tables of values provide numerical evidence which

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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 12 1 f using our work from the synthetic division to write f (x) = x − 2 x + 3 12x2 − 12x + 12 , 2 or f (x) = 12 x − 1 x + 1 x2 − x + 1 . Since the zeros of x2 − x + 1 are nonreal, we call 2 3 2 − x + 1 an irreducible quadratic meaning it is impossible to break it down any further using x 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 f (x)...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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