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PP Section 12.4

# 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.
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
Comments The factorization described in the Linear Factor Theorem is happening over the set of complex numbers. The factors are not necessarily different. The multiplicity of each factor is taken into account in the statement of the theorem.

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Examples 1. Find a polynomial f(x) with leading coefficient 4 such that the equation

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

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