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Denominator: Linear and different factors Worked example Type 1.

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1 Denominator: Linear and different factors Worked example Type 1

2 Check top line is lower degree Find partial fractions for Factorise bottom line A ( 2x – 3 ) B ( x + 2 ) + Split into commonsense fractions. A and B are constants. Note: top line must be lower degree 5x – 11 ( 2x – 3 )( x + 2 ) 5x – 11 2x 2 + x - 6 5x – 11 2x 2 +x - 6 A( x + 2 ) + B( 2x – 3 ) ( 2x – 3 )( x + 2 ) Combine the fractions = = =

3 A( x + 2 ) + B( 2x – 3 ) ( 2x – 3 )( x + 2 ) = 5x – 11 2x 2 +x - 6 => 5x – 11 = A( x + 2 ) + B( 2x – 3 ) 1: Equate coefficients and solve a pair of simultaneous equations. Now find the unknowns A and B. There are two ways to do this. 2: Substitute clever values of x into the identity i.e. values of x which make terms disappear We will use the second method. Identity

4 Identity: 5x – 11 = A( x + 2 ) + B( 2x – 3 ) Let x = -2( x + 2 = 0 ) -10 – 11 = -7B -21 = -7B B = 3 ( 2x - 3 = 0 ) Let x = 3232 – 11 = A – 22 = 7A -7 = 7A A =-1 ……… x (2) Partial fractions are: 5x – 11 2x 2 +x - 6 ( 2x – 3 ) + = 3 ( x + 2 )

5 Denominator: Linear factor and an irreducible quadratic factor Type 2 Worked example

6 Check top line is lower degree Find partial fractions for Factorise denominator fully A ( x + 1 ) Bx + C ( x ) + Now ready to split into commonsense fractions. A, B and C are constants. Note: top line must be lower degree. 3x – 2 ( x + 1 )( x ) 3x – 2 ( x + 1 )( x ) A( x ) +( Bx + C )( x + 1 ) ( x + 1 )( x ) Combine the fractions = = = 3x – 2 ( x + 1 )( x )

7 Identity: 3x – 2 = A( x ) +( Bx + C )( x + 1 ) Let x = -1( x + 1 = 0 ) -3 – 2 = A((-1) 2 + 4) -5 = 5A A = -1 Let x = 0 –2 = 4A + C -2 = -4 + C C = 2 Partial fractions are: We have run out of clever values. Now choose simple values, not already used, and use the values of the constants already found Let x = 1 1 = 5A + ( B + C )( 2 ) 1 = B + 4 2B = 2 B = 1 (but A = -1 and C = 2) = 3x – 2 ( x + 1 )( x ) ( x + 1 ) x + 2 ( x ) + (but A = -1)

8 Denominator: Repeated linear factors Type 3 Worked example

9 Find partial fractions for 7x 2 -11x - 5 ( x + 2 )( x - 1 ) 2 The factor ( x – 1 ) is repeated. There are 3 factors on the bottom line so we must split it into 3 bits. The correct way to split is as follows A ( x + 2 ) B ( x - 1 ) + = 7x 2 -11x - 5 ( x + 2 )( x - 1 ) 2 + C ( x - 1 ) 2 Identity: 7x 2 -11x - 5 = A( x - 1 ) 2 + B( x - 1 )( x + 2 ) + C ( x + 2 ) clever values Let x = 1( x - 1 = 0 ) 7 – = C( 3 ) -9 = 3C C = -3 Let x = -2( x + 2 = 0 ) = A( -3 ) 2 45 = 9A A = 5

10 We have run out of clever values. So choose simple values, not already used, and use the values of the constants already found. Let x = 0 –5 = A(-1) 2 +B(-1)(2) + C(2) -5 = A – 2B + 2C ( but A = 5 and C = -3 ) -5 = 5 – 2B - 6 2B = 4 B = 2 Identity: 7x 2 -11x - 5 = A( x - 1 ) 2 + B( x - 1 )( x + 2 ) + C ( x + 2 ) Partial fractions are: 7x 2 -11x - 5 ( x + 2 )( x - 1 ) 2 5 ( x + 2 ) 2 ( x - 1 ) + = - 3 ( x - 1 ) 2

11 3x 2 + 2x + 9 ( x - 3 )( x - 2 ) 3 4 factors, so split into 4 bits A (x - 3 ) B ( x - 2 ) + = + C ( x - 2 ) 2 + D ( x - 2 ) 3 5x + 7 x 2 ( x + 1 ) x is repeated. 3 factors, so split into 3 bits A x + = + C ( x + 1 ) B x 2 f(x) ( x + a ) n n factors, so split into n bits In general A 1 (x + a ) + = + + A 2 (x + a ) 2 A 3 (x + a ) 3 A n (x + a ) n

12 Numerator same degree or higher than the denominator Worked example

13 Find partial fractions for x( x + 3 ) x 2 + x - 12 Numerator is degree 2, denominator is degree 2. Same degree => DIVIDE OUT first x 2 + x - 12x 2 + 3x x 2 + x x + 12 REMAINDER QUOTIENT => x( x + 3 ) x 2 + x - 12 = x + 12 x 2 + x - 12 NOW FIND PARTIAL FRACTIONS FOR THIS BIT THE FINAL ANSWER IS x( x + 3 ) x 2 + x - 12 = ( x – 3 ) - 4 7( x + 4 )

14 Lets do some more examples Express in partial fractions. 3x 2 + 2x + 1 ( x + 1 )(x 2 + 2x + 2) We must first use the discriminant to verify that the quadratic factor ( x 2 + 2x + 2) is irreducible. For x 2 + 2x + 2 : a =, b =, c = b 2 – 4ac =2 2 – 4(1)(2) -4 b 2 – 4ac < 0 => x 2 + 2x + 2 is irreducible 3x 2 + 2x + 1 ( x + 1 )(x 2 + 2x + 2) = =

15 Another example: Expressas the sum of a polynomial and partial fractions. x 3 - 3x (x 2 - x - 2)


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