• Skip to main content
  • Skip to secondary menu
  • Skip to primary sidebar
  • Skip to footer
  • ICSE Solutions
    • ICSE Solutions for Class 10
    • ICSE Solutions for Class 9
    • ICSE Solutions for Class 8
    • ICSE Solutions for Class 7
    • ICSE Solutions for Class 6
  • Selina Solutions
  • ML Aggarwal Solutions
  • ISC & ICSE Papers
    • ICSE Previous Year Question Papers Class 10
    • ISC Previous Year Question Papers
    • ICSE Specimen Paper 2021-2022 Class 10 Solved
    • ICSE Specimen Papers 2020 for Class 9
    • ISC Specimen Papers 2020 for Class 12
    • ISC Specimen Papers 2020 for Class 11
    • ICSE Time Table 2020 Class 10
    • ISC Time Table 2020 Class 12
  • Maths
    • Merit Batch

A Plus Topper

Improve your Grades

  • CBSE Sample Papers
  • HSSLive
    • HSSLive Plus Two
    • HSSLive Plus One
    • Kerala SSLC
  • Exams
  • NCERT Solutions for Class 10 Maths
  • NIOS
  • Chemistry
  • Physics
  • ICSE Books

Partial Fractions

December 10, 2020 by Prasanna

Partial Fractions

Partial Fractions 1

An expression of the form \(\frac { f(x) }{ g(x) }\), where f(x) and g(x) are polynomial in x, is called a rational fraction.

  1. Proper rational functions: Functions of the form \(\frac { f(x) }{ g(x) }\), where f(x) and g(x) are polynomials and g(x) ≠ 0, are called rational functions of x.
    If degree of f(x) is less than degree of g(x),then is called a proper rational function.
  2. Improper rational functions: If degree of f(x) is greater than or equal to degree of g(x), then \(\frac { f(x) }{ g(x) }\) is called an improper rational function.
  3. Partial fractions: Any proper rational function can be broken up into a group of different rational fractions, each having a simple factor of the denominator of the original rational function. Each such fraction is called a partial fraction.

If by some process, we can break a given rational function \(\frac { f(x) }{ g(x) }\) into different fractions, whose denominators are the factors of g(x),then the process of obtaining them is called the resolution or decomposition of \(\frac { f(x) }{ g(x) }\) into its partial fractions.

Different cases of partial fractions

(1) When the denominator consists of non-repeated linear factors:
To each linear factor (x – a) occurring once in the denominator of a proper fraction, there corresponds a single partial fraction of the form \(\frac { A }{ x-a }\), where A is a constant to be determined.
If g(x) = (x – a1)(x – a2)(x – a3) ……. (x – an), then we assume that,
Partial Fractions 2
where A1, A2, A3, ………. An are constants, can be determined by equating the numerator of L.H.S. to the numerator of R.H.S. (after L.C.M.) and substituting x = a1, a2,…… an.
(2) When the denominator consists of linear factors, some repeated:
To each linear factor (x – a) occurring r times in the denominator of a proper rational function, there corresponds a sum of r partial fractions.
Let g(x) = (x – a)k(x – a1)(x – a2) ……. (x – ar). Then we assume that
Partial Fractions 3
Where A1, A2, A3, ………. Ak are constants. To determine the value of constants adopt the procedure as above.
(3) When the denominator consists of non-repeated quadratic factors:
To each irreducible non repeated quadratic factor ax2 + bx + c, there corresponds a partial fraction of the form \(\frac { Ax+B }{ { a{ x }^{ 2 }+bx+c } }\), where A and B are constants to be determined.
Example :
Partial Fractions 4
(4) When the denominator consists of repeated quadratic factors:
To each irreducible quadratic factor ax2 + bx + c occurring r times in the denominator of a proper rational fraction there corresponds a sum of r partial fractions of the form.
Partial Fractions 5
where, A’s and B’s are constants to be determined.

Partial fractions of improper rational functions

If degree of is greater than or equal to degree of g(x), then \(\frac { f(x) }{ g(x) }\) is called an improper rational function and every rational function can be transformed to a proper rational function by dividing the numerator by the denominator.
We divide the numerator by denominator until a remainder is obtained which is of lower degree than the denominator.

General method of finding out the constants

  1. Express the given fraction into its partial fractions in accordance with the rules written above.
  2. Then multiply both sides by the denominator of the given fraction and you will get an identity which will hold for all values of x.
  3. Equate the coefficients of like powers of x in the resulting identity and solve the equations so obtained simultaneously to find the various constant is short method. Sometimes, we substitute particular values of the variable x in the identity obtained after clearing of fractions to find some or all the constants. For non-repeated linear factors, the values of x used as those for which the denominator of the corresponding partial fractions become zero.

Filed Under: Mathematics Tagged With: Different cases of partial fractions, General method of finding out the constants, Improper rational functions, Partial Fractions, Partial fractions of improper rational functions, Proper rational functions

Primary Sidebar

  • MCQ Questions
  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • ICSE Solutions
  • Selina ICSE Solutions
  • Concise Mathematics Class 10 ICSE Solutions
  • Concise Physics Class 10 ICSE Solutions
  • Concise Chemistry Class 10 ICSE Solutions
  • Concise Biology Class 10 ICSE Solutions
  • Concise Mathematics Class 9 ICSE Solutions
  • Concise Physics Class 9 ICSE Solutions
  • Concise Chemistry Class 9 ICSE Solutions
  • Concise Biology Class 9 ICSE Solutions
  • ML Aggarwal Solutions
  • ML Aggarwal Class 10 Solutions
  • ML Aggarwal Class 9 Solutions
  • ML Aggarwal Class 8 Solutions
  • ML Aggarwal Class 7 Solutions
  • ML Aggarwal Class 6 Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC

Recent Posts

  • Air Pollution Essay for Students and Kids in English
  • 10 Lines on Satya Nadella for Students and Children in English
  • Essay on Wonders of Science | Wonders of Science for Students and Children in English
  • My Childhood Memories Essay | Essay on My Childhood Memories for Students and Children in English
  • Essay On Sports And Games | Sports And Games Essay for Students and Children in English
  • Clean India Slogans | Unique and Catchy Clean India Slogans in English
  • Animals that Start with Z | Listed with Pictures, List of Animals Starting with Z & Interesting Facts
  • Positive Words That Start With S | List of Positive Words Starting With S Meaning, Examples,Pictures and Facts
  • Positive Words that Start with B | List of 60 Positive Words Starting with B Pictures and Facts
  • Humanity Essay | Essay on Humanity for Students and Children in English
  • Teenage Pregnancy Essay | Essay on Teenage Pregnancy for Students and Children in English

Footer

  • RS Aggarwal Solutions
  • RS Aggarwal Solutions Class 10
  • RS Aggarwal Solutions Class 9
  • RS Aggarwal Solutions Class 8
  • RS Aggarwal Solutions Class 7
  • RS Aggarwal Solutions Class 6
  • Picture Dictionary
  • English Speech
  • ICSE Solutions
  • Selina ICSE Solutions
  • ML Aggarwal Solutions
  • HSSLive Plus One
  • HSSLive Plus Two
  • Kerala SSLC
  • Distance Education
DisclaimerPrivacy Policy
Area Volume Calculator