PARTIAL FRACTIONS CUBIC DENOMINATOR

Example 1 :

Resolve the following rational expressions into partial fractions.

x/(x-1)³

Solution :

By decomposing the denominator, we get

x/(x-1)³  =  (A(x - 1)² + B(x - 1) + C)/(x-1)³

x  =  A(x - 1)² + B(x - 1) + C

(1) + (2)

A - B + A + B  =  -1 + 1

2A  =  0

A  =  0

Applying the values of A in the first equation, we get

0 - B  =  -1

B  =  1

Hence the solution is

Example 2 :

Resolve the following rational expressions into partial fractions.

1/(x⁴ - 1)

Solution :

By decomposing the denominator, we get

1 = A(x - 1)(x² + 1) + B(x+1) (x²+1) + (Cx+D)(x+1)(x-1)

Hence the solution is

Example 3 :

Resolve the following rational expressions into partial fractions.

(x - 1)²/(x³ + x)

Solution :

By decomposing the denominator, we get

(x - 1)²  =  A(x² + 1) + (Bx + C) x 

(1) + (2)

-2 + 2   =  B + C + B - C

0  =  2B

B  =  0

By applying the value of B in the first equation, we get

C  =  -2

Hence the solution is

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