7.4 Partial Fraction Decomposition

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7.4 Partial Fraction Decomposition
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A rational expression P / Q is called proper if
the degree of the polynomial in the numerator is
less than the degree of the polynomial in the
denominator. Otherwise the rational expression is
termed improper.
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CASE 1 Q has only non-repeated linear factors.
Under the assumption that Q has only non-repeated
linear factors, the polynomial Q has the form
where none of the numbers ai are equal. In this
case, the partial fraction decomposition of P / Q
is of the form
where the numbers Ai are to be determined.
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Write a partial fraction decomposition of
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Solution of this system is A3, B2.
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CASE 2 Q has repeated linear factors.
If the polynomial Q has a repeated factor, say,
(x - a) n, n gt 2 an integer, then, in the partial
fraction decomposition of P / Q, we allow for the
terms
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Write the partial fraction decomposition of
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Need to solve
From the last and the first we easily get A and
B.
A-3/4, B3/4, C7/2.
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CASE 3 Q contains a non-repeated irreducible
quadratic factor.
If Q contains a non-repeated irreducible
quadratic factor of the form ax2 bx c, then,
in the partial fraction decomposition of P / Q,
allow for the term
where the numbers A and B are to be determined.
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Write a partial fraction decomposition of

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Collect like terms
Comparing right and left sides gives
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(1)
(2)
(3)
(4)
Adding (1) and (3) gives
Adding (2) and (4) gives
2A2B 4
2A-2B -8
Combining these
4A -4
Solution is A -1, B 3, C 2, D 1.
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CASE 4 Q contains repeated irreducible
quadratic factors.
If the polynomial Q contains a repeated
irreducible quadratic fator (ax2 bx c)n, n gt
2, n an integer, then, in the partial fraction
decomposition of P / Q, allow for the terms
where the numbers Ai ,Bi are to be determined.
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Write the partial fraction decomposition of
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Comparing left and right sides leads to