O-Level A-Math and SEC G3 Additional Mathematics K341
A4: Polynomials and partial fractions
Use factor and remainder relationships, polynomial division, and the correct decomposition form.
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Core notes
Polynomial methods connect division, factors, roots, identities, and rational expressions. Preserve the polynomial degree during division, use the remainder or factor theorem at a specific input, and choose a partial-fraction numerator that matches each denominator factor.
Polynomial multiplication and division
Multiply every term systematically and collect equal powers. For division, arrange both polynomials in descending powers and include zero coefficients for missing terms. The result has the form dividend equals divisor times quotient plus remainder, where the remainder has lower degree than the divisor.
Remainder theorem, factor theorem, and cubic equations
When is divided by , the remainder is . Therefore is a factor exactly when . After identifying a linear factor of a cubic, divide it out and solve the remaining quadratic.
Sum and difference of cubes
Recognise and . The middle sign in the quadratic factor is opposite for a sum and positive for a difference.
Partial fractions
If the numerator degree is at least the denominator degree, divide first. For distinct linear factors, use one constant numerator over each factor. A repeated factor needs a term for every power up to its multiplicity. An irreducible quadratic factor needs a linear numerator. Clear denominators, then compare coefficients or substitute convenient values.
Formulae and methods
| Relationship | Use |
|---|---|
| Apply the remainder theorem. | |
| Factor a sum or difference of cubes. | |
| Set up a repeated-linear-factor decomposition. | |
| Set up a linear plus irreducible-quadratic decomposition. |
Worked examples
Example 1: Given , show that is a factor and solve .
- Evaluate , so is a factor.
- Divide to obtain .
- Factor the quadratic: .
Answer: .
Example 2: Resolve into partial fractions.
- Write .
- Clear denominators: .
- Use to get , and to get .
Answer: .
Chapter checkpoint
- Write missing polynomial powers with zero coefficients before division.
- Use to test a proposed factor or remainder and then reduce the polynomial degree.
- Match the decomposition form to distinct, repeated, or irreducible quadratic factors before solving coefficients.
Exam traps and retrieval check
Avoid these traps
- Omitting a zero coefficient for a missing power during polynomial division.
- Using without matching the factor .
- Giving a constant numerator above an irreducible quadratic factor.
Check from memory
What is the remainder when is divided by ?
.
How do you finish solving a cubic after finding one root?
Divide out the corresponding linear factor and solve the remaining quadratic.
What terms are required for a denominator factor ?
Both and .
Official K341 coverage
This chapter maps to 4 official outcomes: A4.1, A4.2, A4.3, A4.4. Check the official 2027 K341 syllabus for exact assessable wording and paper details.