Explanation Given Information: The provided fraction is: − 3 x 2 + 2 x + 8 x 3 + 4 x 2 + 4 x Formula used: For any rational expression f ( x ) g ( x ) , where f ( x ) and g ( x ) area polynomials with real coefficients, degree of f ( x ) is less than the degree of g ( x ) and g ( x ) ≠ 0 . Part I: Step1: Factorize the denominator g ( x ) completely into quadratic factors of the form ( a x 2 + b x + c ) n and linear factors of the form ( a x + b ) m that cannot be factorized further. Step2: Write the original rational expression f ( x ) g ( x ) as a sum of simpler fractions to set up the form for the decomposition. Now for linear factors of g ( x ) : The partial fraction decomposition must include the sum: A 1 ( a x + b ) 1 + A 2 ( a x + b ) 2 + ..... + A m ( a x + b ) m where, A 1 , A 2 , ....... , A m are constants. For quadratic factors of g ( x ) : The partial fraction decomposition must include the sum: B 1 x + C 1 ( a x 2 + b x + c ) 1 + B 2 x + C 2 ( a x 2 + b x + c ) 2 + ....... + B n x + C n ( a x 2 + b x + c ) n where B 1 , B 2 ....... B n and C 1 , C 2 , ..... C n are constant. Part II: Step3: Multiply both sides of the equation by the LCD to clear fractions with the form of the partial fraction decomposition set up, Step4: Use equation from step3 to set up a system of linear equations by equating the constant terms and equating the coefficients of like powers of x . Step5: Now solve the system of equations obtained from step4 and finally, substitute the solutions to the system into the partial fraction decomposition. Calculation: Consider the provided fraction, − 3 x 2 + 2 x + 8 x 3 + 4 x 2 + 4 x Factorize the denominator. − 3 x 2 + 2 x + 8 x 3 + 4 x 2 + 4 x = − 3 x 2 + 2 x + 8 x ( x 2 + 4 x + 4 ) = − 3 x 2 + 2 x + 8 x ( x + 2 ) 2 The denominator has a linear factor of x 1 and a repeated linear factor ( x + 2 ) 2 . Now, for a repeated factor that occurs m times, one fraction must be given for each power less than or equal to m . Since, linear factor is raised to the first power, only one fraction is needed for each factor. Thus, − 3 x 2 + 2 x + 8 x ( x + 2 ) 2 = A x + B ( x + 2 ) + C ( x + 2 ) 2 To solve for A and B , first multiply both sides by the LCD x ( x + 2 ) 2

COLLEGE ALGEBRA CUSTOM TEXT WITH ALEKS 3

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ISBN: 9781260695335

Author: Miller

Publisher: MCG

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Chapter 5.3, Problem 28PE

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To calculate: The partial fraction decomposition for the provided fraction −3x2+2x+8x3+4x2+4x.

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Chapter 5.3, Problem 27PE

Chapter 5.3, Problem 29PE

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COLLEGE ALGEBRA CUSTOM TEXT WITH ALEKS 3

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The partial fraction decomposition for the provided fraction − 3 x 2 + 2 x + 8 x 3 + 4 x 2 + 4 x .

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To calculate: The partial fraction decomposition for the provided fraction −3x2+2x+8x3+4x2+4x.

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