Partial fractions

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Section 6.3 Partial Fractions
Advanced Algebra
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What is Partial Fraction Decomposition?

• There are times when we are working with Rational
  
• Functions of the form when we
  want to
• split it up into two simpler fractions. The
  process that we go through to do this is called
  partial fraction decomposition

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Lets say you had the following
What would you need to do to add them together?
Multiply by a common denominator to both fractions
Write as one fraction and simplify the numerator
The final answer
Partial Fraction Decomposition is the reverse of
what we just did here
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Break into 2 smaller fractions
Multiply by LCD
Cancel anything necessary
Collect like terms and set equal
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Multiply by the common denominator.
Collect like terms together and set them equal to
each other.
Solve two equations with two unknowns.
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This technique is called Partial Fractions
Solve two equations with two unknowns.
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Sometimes you might get a repeated factor
(multiplicity) in the denominator
Repeated roots we must use two terms for partial
fractions.
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Partial-Fraction Decomposition Repeated linear
factor
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Now we have 3 equations with 3 unknowns. Solve
like in previous section.
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If the degree of the numerator is higher than the
degree of the denominator, use long division
first.
(from example one)
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What if the denominator has a non-factorable
quadratic in it?
Partial Fraction Decomposition can get very
complicated, very quickly when there are
non-factorable quadratics and repeated linear
factorshere is an easy example
Now just solve the 3 by 3 system A3, B2 and C4
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A nice shortcut if you have non-repeated linear
factorsthe Heaviside Shortcutnamed after
mathematician Oliver Heaviside (1850-1925)
Tell yourself, if x is 5, then x-5 is 0. Cover
up the x-5 and put 5 in for the x in what is left
Do the same for the other linear factor
Which should probably be simplified
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A challenging example
first degree numerator
irreducible quadratic factor
repeated root
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expand ((-2x4)/((x21)(x-1)2))
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F2
p