Status: Tags: #math/calculus/integrals Links: Integrals of Rational Functions
Partial Fraction Decomposition
Used for turning rational functions with factorable denominators as sum/differences of multiple rational fractions with linear or quadratic denominators
- Opposite of finding a LCD
Steps
- Use polynomial long division if possible
- Completely factor the denominator only
- If not factorable, you must use the other methods
- Use a table to determine the term(s) of use in decomposition for each factor
- Replace factors with PF terms
- A,B, etc, are unknown values
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Denominator Factor Term in PF Linear: ax+b A/ax+b Repeated linear: (ax+b)^2 A/ax+b + B/ax+b… Quadratic: ax^2+bx+c Ax+B/ax^2+bx+c Repeated quad: (ax^2+bx+c)^2 Ax+B/ax^2+bx+c + Cx+D/ax^2+bx+c…
- Set rational function equal to partial fractions
- Remember to add all multiples of the denominator (x-1), (x-1)^2
- Numerator is dependent on the bracket interior
- Remember to add all multiples of the denominator (x-1), (x-1)^2
- Multiply by the denominator to eliminate fractions
- Use “cover-up” method or “expand and compare” method to determine the value of A,B,C, etc
- Partial Fraction Cover-Up Method
- Expand and Compare
- Distribute uppercase variables to factors
- Variables must combine to equal the appropriate coefficients of the initial numerator
- Rewrite rational function as sum/difference of partial fractions using values of A,B,C, etc
Anki
START Cloze Partial Fraction: 2. Set rational function equal to {partial fractions} - $x+a$ is over A - $x^2+a$ is over Ax+B - Remember to add all multiples of the denominator ex) $(x-1), (x-1)^2$ - Numerator is dependent on the {bracket interior} 3. Multiply by {the denominator} to eliminate fractions 4. Solve for a certain variable by choosing an x value that {cancels other factors} 5. If unable to find an x value to isolate a constant, pick an appropriate number and {plug in the values of the known constants} - Can continue testing values to form a linear system Back: DECK: IntegralCalculus Tags: RationalIntegrals
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Practice
References:
Created:: 2021-06-12 14:59