Mathematics · Class 12 Science
Ch 7Partial Fractions — Class 12 Mathematics, concept-first.
The previous chapter worked entirely with polynomial equations — expressions built purely from sums of powers of . But polynomials are just as often divided by one another, as in , and such a quotient is called a rational fraction.
Key concepts
Hover a concept to preview it and jump to its most relevant Q&A.
Partial Fractions
Partial fractions is the technique of breaking a single rational expression f(x)/g(x) back down into a sum of simpler fractions whose denominators are powers of the irreducible pieces of g(x). It reverses the process of…
Most relevant Q&A
- Resolve : $\dfrac{x^3}{(x-1)(x+2)}$ into partial fractions.Preview
- Resolve the following fraction into partial fractions: $\dfrac{x^2 - 3}{(x+2)(x^2+1)}$.Preview
- Resolve $\frac{x^3}{(2x-1)(x-1)^2}$ into partial fractions.Preview
- Resolve $\dfrac{x^2 - x + 1}{(x + 1)(x - 1)^2}$ into partial fractions.Preview
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Introduction
The previous chapter worked entirely with polynomial equations — expressions built purely from sums of powers of .
Rational Fractions, Proper/Improper Fractions and Irreducible Polynomials
Whenever we write one polynomial divided by another, say with , we call the expression a rational fraction (or just a fraction, or a polynomial fraction). For example and are both rational fractions.
Case 1 — g(x) Has Distinct (Non-Repeated) Linear Factors
The rule. Suppose is a proper fraction and factors into linear pieces that are all different from one another, say with no factor repeated.
Case 2 — g(x) Has a Repeated Linear Factor
The rule. If a linear factor occurs times in — that is, divides but does not — then it contributes not one but a whole chain of partial fractions, with every power from up to appearing in a denominato…
Case 3 — g(x) Has a Non-Repeated Irreducible Quadratic Factor
The rule. Sometimes doesn't factor completely into linear pieces — it has a genuinely irreducible quadratic factor , (recall: irreducible means , so it has no real roots and can't be split further).
Case 4 — g(x) Has a Repeated Irreducible Quadratic Factor
The rule. This case combines the ideas of Cases 2 and 3. If an irreducible quadratic occurs times in (i.e.
What to Do First When f(x)/g(x) Is an Improper Fraction
Every rule in the four sections above assumes you're starting from a proper fraction (numerator's degree strictly less than denominator's).
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q1Resolve : $\dfrac{x^3}{(x-1)(x+2)}$ into partial fractions.Preview
- Q2Resolve the following fraction into partial fractions: $\dfrac{x^2 - 3}{(x+2)(x^2+1)}$.Preview
- Q3Resolve $\frac{x^3}{(2x-1)(x-1)^2}$ into partial fractions.Preview
- Q4If $\dfrac{x}{(x-2)(x+3)}=\dfrac{A}{x-2}+\dfrac{B}{x+3}$, find $A$.Preview
- Q5Resolve into partial fractions: $\dfrac{x+4}{(x+1)(x-1)^2}$.Preview
- Q6Resolve into partial fractions: $\dfrac{2x^2+3x+4}{(x-1)(x^2+2)}$.Preview
- Q7Resolve into partial fractions: $\dfrac{x^2+1}{x^2(x-1)}$.Preview
- Q8Resolve $\dfrac{x^2 - x + 1}{(x + 1)(x - 1)^2}$ into partial fractions.Preview