M1410205Part2

M1410205Part2 - 2.57 PART E THE FUNDAMENTAL THEOREM OF...

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2.57 PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA) The Fundamental Theorem of Algebra (FTA) If f x ( ) is a nonconstant n th -degree polynomial in standard form with real coefficients, then it must have at least one complex (possibly real) zero. Put Another Way: It must have exactly n complex zeros, where the zeros may be repeated based on their multiplicities. Technical Note: The Fundamental Theorem of Arithmetic states that any integer greater than or equal to 2 is either prime or can be decomposed uniquely as a product of (possibly repeated) primes (or “prime powers”), up to a reordering of the factors. For example, 6 can only be decomposed in one way: 6 = 2 3 . The decomposition 6 = 3 2 does not count as a different one. Technical Note: The Fundamental Theorem of Calculus will allow you to evaluate definite integrals, which are used in finding areas, volumes, arc lengths, surface areas, and much more. Historical Note: The FTA was first proven by the great Gauss. For more history, see p.193 of the textbook .

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2.58 PART F: THE LINEAR FACTORIZATION THEOREM (LFT) The Linear Factorization Theorem (LFT) If f x ( ) is a nonconstant polynomial in standard form with real coefficients, then it must have a factorization into linear factors of the form: f x ( ) = a n x c 1 ( ) x c 2 ( ) x c n ( ) a n R ; a n 0; each c i C ( ) Note: The zeros of f x ( ) are then c 1 , c 2 , , c n . Note: There may be repetitions of a zero c i , based on the multiplicity of c i . Note: a n is the leading coefficient of f x ( ) . Technical Note: The LFT is proven using the FTA and the Factor Theorem. See p.193 of the textbook . Technical Note: This helps explain the Complex Conjugate Pairs Theorem in Notes 2.56 . Example Let f x ( ) = x 5 8 x 4 + 16 x 3 . x
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This note was uploaded on 09/08/2011 for the course MATH 141 taught by Professor Staff during the Fall '11 term at Mesa CC.

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M1410205Part2 - 2.57 PART E THE FUNDAMENTAL THEOREM OF...

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