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Unformatted text preview: Objectives
Understand the implications of the fundamental theorem of algebra Find ALL the roots to a given polynomial equation Section 6.6
THE FUNDAMENTAL THEOREM OF ALGEBRA Carl Friedrich Gauss
In 1799 Gauss proved that if you solve any polynomial equation, your roots are included in the complex numbers. This seems obvious, but the idea was so important that we call it THE FUNDAMENTAL THEOREM OF ALGEBRA. Theorem If P(x) is a polynomial of degree n 1 with complex coefficients, then P(x) = 0 has at least one complex root Corollary (Including imaginary roots and multiple roots) an nth degree polynomial equation has exactly n roots; the related polynomial function has exactly n zeros. 1 x3 + 2 x 2 - 4 x - 6 = 0
How many complex roots does this equation have? How many imaginary roots could this equation have? How many irrational roots could this equation have? What are possible rational roots for this equation? x3 + x 2 - x + 2 = 0
Find all the roots of the equation. f ( x) = 2 x 3 - 4 x 2 + 8 x - 16
Find all the zeros of this function f ( x) = 2 x 4 + 2 x 3 - 4 x - 4
List the possible rational zeros; then find all the zeros of the function Homework #___
Page 337 (1-15odd), 25 (127 is bonus (+1 on the HW). Turn it in on a note card or a small slip of paper If you're still missing a quiz or a test, now you' would be a good time to make it up! 2 ...
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