MathematicsChapter 3: Integration
Repeated and Irreducible Factors
Repeated linear factors need every power; irreducible quadratics use a linear numerator.
Learning objectives
- State the rule for repeated and irreducible factors.
- Solve a repeated and irreducible factors problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Repeated linear factors need every power; irreducible quadratics use a linear numerator.
Where / symbols
- Repeated and irreducible factors
- Repeated linear factors need every power; irreducible quadratics use a linear numerator.
CONCEPT AND RULE
Repeated linear factors need every power; irreducible quadratics use a linear numerator.
METHOD
For repeated and irreducible factors, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
For 1/[x(x-1)2], include Ax+Bx-1+Cx-12.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve repeated and irreducible factors
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- For 1/[x(x-1)2], include Ax+Bx-1+Cx-12.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Repeated and irreducible factors
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Repeated and Irreducible Factors
Sticky Notes · Formula and Worked Revision
Core Idea
- Repeated linear factors need every power; irreducible quadratics use a linear numerator.
Rule / Formula
- Repeated linear factors need every power; irreducible quadratics use a linear numerator.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- For 1/[x(x-1)2], include Ax+Bx-1+Cx-12.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- Repeated linear factors need every power; irreducible quadratics use a linear numerator.
- Worked pattern: For 1/[x(x-1)^2], include Ax+Bx-1+Cx-12.
- Final check: domain, sign, notation, units, and reasonableness.
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