Partial Fractions
Decompose a rational function into simpler fractions.
Key idea
Decompose a rational function into simpler fractions.
In depth
A rational expression can be split into a sum of simpler fractions with factored denominators. Set up unknown numerators, multiply through, and solve for them. It is crucial for integration.
Worked example
1/(x(x+1)) = A/x + B/(x+1) → A = 1, B = −1.
Formula
Real-world uses
- Integrating rational functions in engineering
- Solving circuit and control system problems
Common mistake
Using partial fractions when the numerator degree is not less than the denominator; divide first.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Partial Fractions?
Decompose a rational function into simpler fractions. It is a topic at the expert level in Algebra.
What is the formula for Partial Fractions?
The key formula is (Ax+B)/((x−p)(x−q)) = α/(x−p)+β/(x−q).
Where is Partial Fractions used in real life?
Real-world examples: Integrating rational functions in engineering; Solving circuit and control system problems.
What should I learn before Partial Fractions?
Make sure you are comfortable with Binomial Theorem, the previous topic in Algebra.
What is a common mistake with Partial Fractions?
Using partial fractions when the numerator degree is not less than the denominator; divide first.
How can I practise Partial Fractions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Partial Fractions?
Move on to Mathematical Induction next.
Related expert topics in Algebra
- Linear Programming – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
- Rational Root Theorem – Possible rational roots are ± factors of constant over factors of leading coefficient.
- Fundamental Theorem of Algebra – Every degree-n polynomial has exactly n complex roots counted with multiplicity.
- Binomial Theorem – Expand (a+b)ⁿ with binomial coefficients.
- Mathematical Induction – Prove statements for all n by base case and inductive step.
- Cramer’s Rule – Solve linear systems using determinants.