math academy-partial-fractions-notes
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Having a positive mental attitude is asking how something can be done rather than saying it can't be done.
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Notes: Partial Fractions Introduction We have learnt how to combine fractions as follows:
)2)(1(12
22
11
−+
−=
−+
+ xxx
xx
Partial Fractions is the reverse process of splitting a single fraction into a sum of two or more fractions, i.e
22
11
)2)(1(12
−+
+=
−+
−
xxxxx
[A] Polynomial
A polynomial in one variable x is given by
012
21
1 ... axaxaxaxa nn
nn +++++ −
−
where 0121 ,,,, aaaaa nn − are constants and n is a non negative integer.
If 0≠na , then the polynomial has degree n .
Egs (i) 5432 24 +++ xxx is a polynomial of degree 4
(ii) 23 16x
x − is not a polynomial
[B] Rational Function
A rational function is an expression of the form )()(xQxP
where )(xP and )(xQ are
polynomials.
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[C] Proper and Improper Function
The rational function )()(xQxP
is a
(i) Proper Fraction if the degree of )(xP < degree of )(xQ
4
3 53xxx −+
(ii) Improper Fraction if the degree of )(xP ≥ degree of )(xQ
2
2 53xxx −+ ,
125
34
6
+− xxx
Case 1: Linear Factor (ax+b) To every linear factor (ax+b) in the denominator of a proper fraction, there corresponds
a partial fraction of the form bax
A+
.
Example 1: Express )3)(2(
12++
+
xxx
in partial fractions.
[3 methods: Substitution, Comparing coefficients and Cover up Rule] ws 1 Q1
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Case 2 : Repeated Linear Factor (ax+b) To every repeated linear factor (ax+b) repeated n times in the denominator of a proper fraction, there corresponds a sum of n partial fractions:
nn
baxA
baxA
baxA
)(...
)( 221
+++
++
+
Example 2: Express 2)32(12
−
−
xx
in partial fractions
ws 1 Q2
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Case 3: Quadratic Factor (ax 2 +bx+c) which cannot be factorised To every quadratic factor (ax 2 +bx+c) in the denominator of a proper fraction, there
corresponds a partial fraction of the form cbxax
BAx++
+2 .
Example 3: Express )2)(12(
152
2
++
+
xxx in partial fractions
ws 1 Q3
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Two Important Checks Before Starting on Any Question Check 1: (Must be a Proper Fraction) Otherwise Apply Long Division First
Example 4: Express 542
3
−− xxx in partial fractions
Check 2: (Denominator Must Be Completely Factorised)
Example 5: Express 65
122 ++
+
xxx
in partial fractions
ws 1 Q4,5 END