Polynomials and partial fractions are foundational A-Math skills that feed directly into solving equations and, at JC level, integration. A solid grasp now saves significant time in examinations.
These notes cover the division algorithm, the Remainder and Factor Theorems, factorising cubics, and expressing proper and improper rational functions in partial fractions for distinct, repeated and irreducible quadratic factors.
What these notes cover
- •Polynomials
- •Division Algorithm
- •Remainder Theorem
- •Factor Theorem
- •Factorising a cubic
- •Proper and improper fractions
- •Case 1 — Distinct linear factors
- •Case 2 — Repeated linear factor
- •Case 3 — Irreducible quadratic factor
- •Worked examples
- •Common mistakes
- •Summary
Key definitions
- Polynomial of degree n
- An expression P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where aₙ ≠ 0, n is a non-negative integer and the aᵢ are real constants.
- Degree and leading coefficient
- The degree is the highest power of x; the leading coefficient is aₙ.
- Proper rational function
- A rational function f(x) ÷ g(x) in which the degree of f is strictly less than the degree of g.
- Improper rational function
- A rational function f(x) ÷ g(x) in which the degree of f is greater than or equal to the degree of g; it must be long-divided before decomposing.
- Irreducible quadratic factor
- A quadratic with b² − 4ac < 0, which has no real roots; in partial fractions it takes a linear numerator Ax + B.
Key formulas
Division Algorithm
P(x) = D(x)Q(x) + R(x)
Dividing P(x) by a divisor D(x) of lower degree gives a unique quotient Q(x) and remainder R(x). The degree of R is strictly less than the degree of D; when dividing by a linear factor, R is a constant.
Remainder Theorem
P(x) is the polynomial being divided. The remainder is found by substitution — no long division required.
Factor Theorem
(x − a) is a factor of P(x) if and only if P(a) = 0
To factorise a cubic: try x = ±1, ±2, … (factors of the constant term) to find a root x = a, divide P(x) by (x − a) to obtain a quadratic, then factorise the quadratic if possible.
Improper fraction — divide first
Use when the fraction is improper. S(x) is the polynomial part from long division and R(x) ÷ g(x) is now proper; decompose that part.
Case 1 — Distinct linear factors
After multiplying through, substitute x = −b ÷ a and x = −d ÷ c to find A and B directly.
Case 2 — Repeated linear factor
Write one term for each power of the repeated factor up to the highest.
Worked example
Express (x² + 3x + 1) ÷ [(x + 1)²(x + 2)] in partial fractions.
- 1The denominator has a repeated linear factor, so write (x² + 3x + 1) ÷ [(x + 1)²(x + 2)] = A ÷ (x + 1) + B ÷ (x + 1)² + C ÷ (x + 2).
- 2Multiply through by (x + 1)²(x + 2): x² + 3x + 1 = A(x + 1)(x + 2) + B(x + 2) + C(x + 1)².
- 3Let x = −1: 1 − 3 + 1 = B(1), so B = −1.
- 4Let x = −2: 4 − 6 + 1 = C(1), so C = −1.
- 5Compare coefficients of x²: 1 = A + C, so A = 2.
- 6∴ (x² + 3x + 1) ÷ [(x + 1)²(x + 2)] = 2 ÷ (x + 1) − 1 ÷ (x + 1)² − 1 ÷ (x + 2).
Watch out: Do not omit the B ÷ (x + 1)² term — a repeated factor needs a separate term for every power up to the highest.
Common mistakes
Decomposing an improper fraction directly
Wrong: Splitting into partial fractions when the numerator degree is greater than or equal to the denominator degree.
Right: Check proper vs improper first; if improper, long-divide before decomposing.
Constant numerator over an irreducible quadratic
Wrong: Writing just a constant A over a quadratic factor with b² − 4ac < 0.
Right: Use a linear numerator: write Ax + B on top.
Wrong sign in the Remainder Theorem
Wrong: Substituting x = 3 when dividing by (x + 3).
Right: Dividing by (x + 3) means substituting x = −3.
Questions students ask
How do I find the remainder of a polynomial without long division?
By the Remainder Theorem, the remainder when P(x) is divided by (x − a) is simply P(a). More generally, dividing by (ax + b) gives remainder P(−b ÷ a).
How do I factorise a cubic polynomial in A-Math?
Try x = ±1, ±2, … (factors of the constant term) until P(a) = 0, which by the Factor Theorem means (x − a) is a factor. Divide P(x) by (x − a) using long division to obtain a quadratic, then factorise the quadratic if possible.
What is the difference between proper and improper fractions in partial fractions?
A rational function f(x) ÷ g(x) is proper when the degree of f is strictly less than the degree of g, and improper otherwise. If it is improper, perform polynomial long division first, then decompose the proper part.
What partial fraction form should I use for each type of denominator?
Use constant numerators over linear factors, linear numerators over irreducible quadratics, and one term per power for repeated factors. Always verify by recombining the partial fractions over a common denominator.
In the full PDF
- •Worked example on the Remainder and Factor Theorems: finding a remainder and fully factorising P(x) = x³ − 2x² − 5x + 6
- •Worked example on an improper fraction: long division first, then decomposing the proper part
- •Case 3 — the full partial fraction form for an irreducible quadratic factor
- •The remaining common mistakes, including how to verify an answer by recombining
- •End-of-topic summary of every rule