6.6notes for mon feb 8

6.6notes for mon feb 8 - Carl Friedrich Gauss In 1799 Gauss...

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Unformatted text preview: 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 iimportant that we so mportant call it THE FUNDAMENTAL THEOREM OF ALGEBRA. THEOREM 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. x3 + 2 x 2 − 4 x − 6 = 0 How many complex roots does this equation have? have? How many imaginary roots could this equation could this have? have? How many irrational roots could this equation have? have? W hat are possible rational roots for this equation? equation? x3 + 2 x 2 − 4 x − 6 = 0 Find all the roots of the equation. Find f ( x) = x 3 + x 2 − x + 2 Find all the zeros of this function Find 1 ...
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