write the Partial fraction decomposition of

Question

x/(x+2)(x+3)

Answer 1

x / [(x + 2)(x + 3)] = a/(x + 2) + b/(x + 3)

 

Get a common denominator.

 

x / [(x + 2)(x + 3)] = [a(x + 3) + b(x + 2)] / [(x + 2)(x + 3)]

 

Equate numerators.

 

x = a(x + 3) + b(x + 2)

 

Expand right hand side and collect coefficients.

 

x = (a + b)x + (3a + 2b)

 

Equate coefficients of like terms across left and right sides.

 

a + b = 1

3a + 2b = 0

 

Solve for a and b.

 

a = -2

b = 3

 

x / [(x + 2)(x + 3)] = -2/(x + 2) + 3/(x + 3)