Mathematics · Class 12 Science
Ch 10Partial Fractions — Class 12 Mathematics, concept-first.
A rational fraction is a quotient of two polynomials, , with . It is proper when the numerator's degree is less than the denominator's, and improper when the numerator's degree is greater than or equal to the denominator's.
Key concepts
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Partial Fractions with Non-Repeated Linear Factors
When the denominator of a proper fraction factors into distinct linear factors, , the decomposition assigns one constant to each factor: Multiplying both sides by clears all denominators and produces a polynomial identit…
Most relevant Q&A
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Rational Fractions: Proper and Improper
A rational fraction is a quotient of two polynomials, , with . It is proper when the numerator's degree is less than the denominator's, and improper when the numerator's degree is greater than or equa…
Non-Repeated Linear Factors
When the denominator of a proper fraction is a product of distinct linear factors, , assign one unknown constant per factor: Multiplying both sides by clears every denominator and leaves a polynomial…
Repeated Linear Factors
A repeated linear factor in the denominator cannot be given a single constant numerator -- the correct decomposition needs an entire descending chain of powers: After clearing denominators, substituti…
Irreducible Quadratic Factors
A quadratic factor of the denominator is called irreducible over the reals when it has no real linear factors, i.e. .
+−Exercise 7(a)i6 questions
- Q1Resolve $\dfrac{3x-1}{(x-1)(x-2)}$ into partial fractions.Free
- Q2Resolve $\dfrac{x^2+1}{(x-1)(x-2)(x-3)}$ into partial fractions.Free
- Q3Resolve $\dfrac{x^2+1}{(x-1)^3}$ into partial fractions.Preview
- Q4Resolve $\dfrac{x^2+x+1}{(x-1)^2(x+2)}$ into partial fractions.Preview
- Q5Resolve $\dfrac{x+3}{(x-1)(x^2+1)}$ into partial fractions.Preview
- Q6Resolve $\dfrac{x^3}{(x-1)(x-2)}$ into partial fractions.Preview
Power Series Expansion Using Partial Fractions
The partial-fraction decomposition of a rational function is not only an algebraic simplification -- it is also the fastest route to its power-series expansion.
+−Exercise 7(d)i4 questions
- Q1Find the coefficient of $x^n$ in the expansion of $\dfrac1{(1-x)(1-2x)}$ as a series of ascending powers of $x$, stating the range of validi…Free
- Q2Find the coefficient of $x^n$ in the expansion of $\dfrac{x}{(1-x)^2(1-2x)}$ as a series of ascending powers of $x$, stating the range of va…Free
- Q3Find the coefficient of $x^3$ in the expansion of $\dfrac{2x+3}{(1-x)(2+x)}$ as a series of ascending powers of $x$, stating the range of va…Preview
- Q4Find the coefficient of $x^4$ in the expansion of $\dfrac{x^2}{(1-x)(1-x^2)}$ as a series of ascending powers of $x$, stating the range of v…Preview
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
+−Show 6 questionsHide questions6 questions
- Q1Resolve $\dfrac{2x^2 + 3x + 4}{(x-1)(x^2+2)}$ into partial fraction.Preview
- Q2Resolve $\dfrac{3x-1}{(1-x+x^2)(x+2)}$ into partial fraction.Preview
- Q3Resolve the following fraction into partial fractions: $\dfrac{5x+6}{(2+x)(1-x)}$.Preview
- Q4Resolve $\dfrac{5x + 1}{(x + 2)(x - 1)}$ into partial fractions.Preview
- Q5Resolve $\frac{x^3}{(x-a)(x-b)(x-c)}$ into partial fractions.Preview
- Q6Resolve $\dfrac{x^2 + 5x + 7}{(x - 3)^3}$ into partial fractions.Preview