Partial Fractions

Partial Fractions

1. Rational fraction or Polynomial fraction or fraction

   • f(x),g(x) are two polynomials,g(x) is non-zero,then f(x)/g(x) is called

2. Proper and Improper Fraction

   • f(x),g(x) are two polynomials,if degre of f(x) is greater than g(x) f(x)/g(x) is called improper or else

3. Irreducible Polynomial

   • Every linear polynomial is irreducible
   • if a=/0,ax2+bx+c,iff b2-4ac<0

4. Division Algorithms

   • g(x) is =/ 0 there exists unique polynomials q(x) and r(x) such that f(x)=q(x)g(x)+r(x)
     • where either r(x)=0 or the degree of r(x) is less than the degree of g(x)

5. Rule 1

   • Partial fractions of f(x)/g(x), when g(x) contains non-repeated linear factors

6. Rule 2

   • Partial fractions of f(x)/g(x), when g(x) contains repeated and / or non-repeated linear factors

7. Rule 3

   • Partial fractions of f(x)/g(x), when g(x) contains irreducible factors

8. Rule 4

9. Partial fractions of f(x)/g(x), when f(x)/g(x) is an improper fraction

   • Case 1
     • If the degree of f(x) =the degree of g(x) then by division algorithm there exists a unique constant k and polynomial r(x) such that f(x)/g(x)=k.g(x)/g(x)+r(x)/g(x)
   • Case 2
     • if f(x)/g(x) is an improper fraction with degree of f(x) > the degree of g(x),then it can be expressed as f(x)/g(x)=q(x)+r(x)/g(x)