PP Section 12.4 - The Fundamental Theorem of Algebra...

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The Fundamental Theorem of Algebra
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Motivation Over the set of real numbers not all polynomial equations have a root. Example: x 2 + 1 = 0 has no real solution. We want to determine the conditions that guarantee that every polynomial equation has a root. Equivalently, we want to know when a polynomial factors completely as a product of linear factors.
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Fundamental Theorem of Algebra Every polynomial equation f ( x ) = 0 of degree n > 1 has at least one root within the complex number system. Recall that real numbers are complex numbers, so the root mentioned in the theorem may be a real number.
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Linear Factor Theorem Every polynomial function f ( x ) of degree n > 1 can be factored into n linear factors. f x a x r x r x r n n ( ) ( )( ) ( ) = - - ⋅ ⋅ - 1 2 f x a x a x a x a n n n n ( ) = + + + + - - 1 1 1 0 That is, if then
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The factorization described in the Linear Factor Theorem is happening over the set of complex numbers. The factors are not necessarily different.
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PP Section 12.4 - The Fundamental Theorem of Algebra...

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