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Unformatted text preview: Lecture 16, Part I: Section 2.5 Zeros of Polynomial Functions Linear Factorization Theorem Every polynomial function u1D453 ( u1D465 ) of degree u1D45B > 0 can be factored into u1D45B linear factors (not necessarily distinct) of the form u1D453 ( u1D465 ) = u1D44E u1D45B ( u1D465 − u1D450 1 )( u1D465 − u1D450 2 ) ⋅ ⋅ ⋅ ( u1D465 − u1D450 u1D45B ) where u1D450 1 , u1D450 2 , . . . , u1D450 u1D45B are complex numbers. That is, every polynomial function of degree u1D45B > has exactly n (not necessarily distinct) zeros in the complex number system. ex. Find all zeros of u1D453 ( u1D465 ) = u1D465 4 − 16 in the complex number system. The Rational Zero Test If the polynomial u1D453 ( u1D465 ) = u1D44E u1D45B u1D465 u1D45B + u1D44E u1D45B − 1 u1D465 u1D45B − 1 + ⋅ ⋅ ⋅ + u1D44E 1 u1D465 + u1D44E has integer coeﬃcients, then every rational zero of u1D453 has the form u1D45D u1D45E (in lowest terms), where u1D45D is a factor of u1D44E and u1D45E is a factor of u1D44E u1D45B...
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