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Mathematics · Class 10 · Chapter 21: Partial Fractions

8 min

Key points

Partial fractions is a method of expressing a rational fraction as a sum of simpler fractions
The degree of numerator must be less than the degree of denominator for partial fraction decomposition
If degree of numerator ≥ degree of denominator, first perform polynomial long division
For linear factors (ax + b), partial fraction has form A/(ax + b)
For repeated linear factors (ax + b)², use A/(ax + b) + B/(ax + b)²
For quadratic factors (ax² + bx + c), partial fraction has form (Ax + B)/(ax² + bx + c)
Always find LCM to combine fractions and equate coefficients to solve for constants
Method of substitution can be used when suitable values make calculation easier

Definitions

Partial Fractions: A method of expressing a proper rational fraction as a sum of two or more simpler fractions whose denominators are factors of the original denominator
Proper Rational Fraction: A rational fraction in which the degree of the numerator is less than the degree of the denominator
Improper Rational Fraction: A rational fraction in which the degree of the numerator is greater than or equal to the degree of the denominator

Worked example

Resolve (5x+7)/((x+1)(x+3)) into partial fractions. (4 marks)

1/(x+1) + 4/(x+3)

Common mistakes

Students forget to check if the fraction is proper before decomposing
Writing wrong partial fraction form for repeated factors
Making calculation errors when substituting values to find constants
Not finding common denominator correctly when combining fractions
Forgetting to factorize denominator completely before starting decomposition

Quick recap

Partial fractions decompose rational functions into simpler fractions. For linear factors (x+a)(x+b), use A/(x+a) + B/(x+b) form. Clear fractions by multiplying through, then substitute strategic x-values to find constants A and B. Always verify by adding partial fractions back to original expression.

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