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Math academy-partial-fractions-notes

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Math academy-partial-fractions-notes

  1. 1. 4047 AM ©Jolyn Ang @ www.MathAcademy.sg 1 Having a positive mental attitude is asking how something can be done rather than saying it can't be done. Bo Bennett quotes Notes: Partial Fractions Introduction We have learnt how to combine fractions as follows: )2)(1( 12 2 2 1 1 −+ − = − + + xx x xx Partial Fractions is the reverse process of splitting a single fraction into a sum of two or more fractions, i.e 2 2 1 1 )2)(1( 12 − + + = −+ − xxxx x [A] Polynomial A polynomial in one variable x is given by 01 2 2 1 1 ... axaxaxaxa n n n n +++++ − − 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) 2 3 1 6 x x − is not a polynomial [B] Rational Function A rational function is an expression of the form )( )( xQ xP where )(xP and )(xQ are polynomials.
  2. 2. 4047 AM ©Jolyn Ang @ www.MathAcademy.sg 2 [C] Proper and Improper Function The rational function )( )( xQ xP is a (i) Proper Fraction if the degree of )(xP < degree of )(xQ 4 3 53 x xx −+ (ii) Improper Fraction if the degree of )(xP ≥ degree of )(xQ 2 2 53 x xx −+ , 12 5 34 6 +− xx x 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 ++ + xx x in partial fractions. [3 methods: Substitution, Comparing coefficients and Cover up Rule] ws 1 Q1
  3. 3. 4047 AM ©Jolyn Ang @ www.MathAcademy.sg 3 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: n n bax A bax A bax A )( ... )( 2 21 + ++ + + + Example 2: Express 2 )32( 12 − − x x in partial fractions ws 1 Q2
  4. 4. 4047 AM ©Jolyn Ang @ www.MathAcademy.sg 4 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( 15 2 2 ++ + xx x in partial fractions ws 1 Q3
  5. 5. 4047 AM ©Jolyn Ang @ www.MathAcademy.sg 5 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 −− xx x in partial fractions Check 2: (Denominator Must Be Completely Factorised) Example 5: Express 65 12 2 ++ + xx x in partial fractions ws 1 Q4,5 END

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