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Tutorial on decomposing complicated fractions into simpler manageable fractions. One of its important applications is in Integration Using Partial Fractions in calculus.
Rules of Decomposition
Example Let f(x) = (2x - 1) / (x3 + 2 x2 + 4x) The denominator is factored as follows (x3 + 2 x2 + 4x) = x(x2 + 2 x + 4) The quadratic term x2 + 2 x + 4 is irreducible over the reals. 2 - For each factor of the form (ax + b)m, the decomposition includes the following sum of fractions C1 / (ax + b) + C2 / (ax + b)2 + ... + Cm / (ax + b)m Example The fraction 2 / (x - 2)3 is decomposed as 2 / (x-2)3 = C1 / (x - 2) + C2 / (x - 2)2 + C3 / (x - 2)3 3 - For each factor of the form (ax2 + bx + c)n, the decomposition includes the following sum of fractions (A1 x + B1) / (ax2 + bx + c) + (A2 x + B2) / (ax2 + bx + c)2 + ... + (An x + Bn) / (ax2 + bx + c)n
Example 1: Decompose into partial fractions
Solution to Example 1:
Multiply both side of the above equation by the least common denominator, (x - 2)(x + 1), and simplify to obtain an equation of the form 2x+5 = A(x + 1) + B(x - 2) Expand the right side and group like terms 2x + 5 = x (A + B) + A - 2B For the right and left polynomials to be equal we need to have 2 = A + B and 5 = A - 2B Solve the above system to obtain A = 3 and B = -1 Substitute A and B in the suggested decomposition above to obtain 
As an exrcise, group terms on the right to obatin the left side
Example 2: Decompose into partial fractions
Solution to Example 2:
Multiply both side of the above equation by (x + 1)2, and simplify to obtain an equation of the form 1 - 2x = A(x + 1) + B Expand the right side and group like terms -2x + 1 = A x + (A + B) For the right and left polynomials to be equal we need to have - 2 = A and 1 = A + B Solve the above system to obtain A = - 2 and B = 3 Substitute A and B in the suggested decomposition above to obtain 
Example 3: Decompose into partial fractions
Solution to Example 3:
Multiply both side of the above equation by (x - 2)(x2 + 2x + 3), and simplify to obtain an equation of the form 4 x2 - x + 8 = A(x2 + 2x + 3) + (Bx + C)(x - 2) The above equality is true for all values of x, let us use x = 2 to obtain an equation in A 22 = 11 A Solve for A to obtain A = 2 In order to find C, we use x = 0 in the above equality 8 = 6 - 2 C Solve for C to obtain C = -1 To find B, we now use x = 1 in the above equality 11 = 12 + (B - 1)(1 - 2) Solve for B to obtain B = 2 The given fraction can be decomposed as follows 
Exercises: Decompose the following fractions into partial fractions.
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