1120_3 - Week 3 We wish Andrew Bynum to recover well and...

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Week 3 We wish Andrew Bynum to recover well and quickly. Special proposal to Kobe Bryant for his 61 points at Madison Square Garden. Outline Method of partial fractions –Case 1. The denominator is a product of distinct linear factors –Case 2. The denominator is a product of linear factors, some of which are repeated –Case 3. The denominator contains distinct irreducible quadratic factors –Case 4. The denominator contains a repeated irreducible quadratic factor How to evaluate coefficients –Clear the fractions and compare the coefficients (most general argument) –Heaviside method (often used in case 1) –Differentiation (often used if denominator has the form ( x - r ) n ) Definition of an Improper Integral –Type I: singularities are infinite –Type II: singularities are finite Decide whether an improper integral converges or diverges –Comparison test –Limit comparison test Goal In the first part our integrands will be the rational functions, which have the form f ( x ) g ( x ) , where f and g are both polynomials. Recall that the degree of a non-zero polynomial f ( x ) is the largest nonnegative integer k such that the coefficient of x k is not zero, denoted by deg( g ). For example, x 4 + 5 x 3 + 4 x + 5 has degree 4. Assumption For the rational function f ( x ) g ( x ) , we assume deg( f ) < deg( g ), otherwise we may apply the long division to express f as f ( x ) = P ( x ) g ( x )+ R ( x ), where P and R are both polynomials such that deg( P ) = deg( f ) - deg( g ) and deg( R ) < deg( g ). That is, f g = P + R g . Fundamental Theorem of Algebra 1
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Any polynomial with real coefficients can be written as a product of real linear factors and real quadratic factors. That is, if f is a polynomial whose coefficient of the term x deg( f ) is 1, then we can write f as f ( x ) = ( x - r 1 ) a 1 · · · ( x - r n ) a n ( x 2 + p 1 x + q 1 ) α 1 · · · ( x 2 + p m x + q m ) α m , where a i and α j are all positive integers.
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