### Type I

**When the denominator is expressible as a product of linear factors**

**Solution:**

Putting x-2=0 or x=2 in (ii), we get 1=3BÂ

Putting x+1 =0 or x=-1 in (ii), we get -2=-3AÂ

Substituting the values of A and B in (i) , we get

### Type II

**When denominator contains some repeating linear factors**

**Solution:**

Putting x-2=0 i.e. x=2 in (ii), we get 7=4 BÂ

Putting x+2 =0 i.e. x=-2 in (ii) we get -5=16 CÂ

Comparing coefficients ofÂ on both sides of the identity (ii) we get

Substituting the values of A,B and C in (i) we get

### Type III

**The denominator contains irreducible quadratic factors**

**Solution:**

Putting x-1 in (ii), we get 1=5 AÂ Putting x=0 in (ii), we get 0= 4 A – C

Putting x =-1 in (ii), we get -1 = 5 A + 2 B -2 C

Substituting the values of A , B and C in (i), we obtain

### Type IV

**If rational function contains only even powers of x in both the numerator and denominator, then to resolve it into partial functions, we proceed as follows:**

**Step I**Â Put in the given rational function

**Step II** Resolve the rational function obtained in step I into partial fractions.

**Step III** Replace y byÂ

**Example**

**Solution:**

Putting y=-1 and y=-4 successively in (ii), we getÂ

Substituting the values of A and B in (i), we obtain

Replacing y byÂ , we obtain