# Algebra and Functions (Edexcel IAL P2)

> Edexcel International A-Level Mathematics · Edexcel IAL Maths P2 (WMA12)
> Source: https://www.owlsprep.com/study/edexcel-ial-math-p2-algebra-and-functions/

This guide covers Edexcel IAL P2 algebraic division by linear divisors, the Factor and Remainder Theorems, and cubic polynomial factorisation, aligned exactly to the 2018 WMA12 specification.

**Prerequisites:** [Knowledge of P1 polynomial basics and index laws](https://www.owlsprep.com/study/edexcel-ial-math-p1-polynomial-basics/); [Ability to expand linear and quadratic expressions](https://www.owlsprep.com/study/edexcel-ial-math-p1-expanding-brackets/)

## Learning objectives

- Perform algebraic division of polynomials by linear divisors of the form $(ax\pm b)$
- Apply the Remainder Theorem to calculate remainders for division by linear expressions
- Use the Factor Theorem to find linear factors and fully factorise cubic polynomials
- Correctly identify and label quotient and remainder terms for polynomial division
- Solve multi-part exam-style problems combining these algebraic concepts

## Algebraic Division by Linear Divisors

**Polynomial Division Rules** — When a degree $n$ polynomial (dividend) is divided by a linear (degree 1) divisor, the result is a degree $n-1$ quotient polynomial plus a constant remainder.

*Example:* Dividing $x^3 + 3x^2 - 4$ by $(x+2)$ gives quotient $x^2 + x - 2$ and remainder 0.

For P2, you only need to divide by linear divisors of the form $(ax\pm b)$. You may use either long division or coefficient matching (equating coefficients method) for this process, both are accepted by Edexcel examiners as long as full working is shown.

**Worked example:** Find the quotient and remainder when $f(x) = 2x^3 + 5x^2 - x - 6$ is divided by $(x + 2)$.

1. Use coefficient matching: let $2x^3 + 5x^2 - x - 6 = (x+2)(Ax^2 + Bx + C) + R$
2. Expand the right-hand side: $Ax^3 + (2A + B)x^2 + (2B + C)x + (2C + R)$
3. Equate coefficients of matching powers of $x$:
4. $$A = 2 \text{ (}x^3 \text{ term)}$$
5. $$2A + B = 5 \implies 4 + B =5 \implies B=1 \text{ (}x^2 \text{ term)}$$
6. $$2B + C = -1 \implies 2 + C = -1 \implies C=-3 \text{ (}x \text{ term)}$$
7. $$2C + R = -6 \implies -6 + R = -6 \implies R=0 \text{ (constant term)}$$
8. State final result: Quotient = $2x^2 + x - 3$, Remainder = 0

> **Exam tip:** Always label quotient and remainder explicitly, as exam mark schemes allocate separate marks for each term.

*Calculator:* allowed

## Remainder Theorem

**Remainder Theorem** — For any polynomial $f(x)$, the remainder when $f(x)$ is divided by $(ax - b)$ is equal to $f\left(\frac{b}{a}\right)$. For divisor $(ax + b)$, the remainder is $f\left(-\frac{b}{a}\right)$.

*Notation:* $f\left(\frac{b}{a}\right)$

> **tip**
>
> The Remainder Theorem eliminates the need for full algebraic division when you only need to find the remainder, saving time in exams.

**Worked example:** Find the remainder when $f(x) = 6x^3 + 11x^2 - x - 6$ is divided by $(2x - 1)$.

1. For divisor $(2x -1)$, $a=2$, $b=1$, so evaluate $f\left(\frac{1}{2}\right)$
2. $$f\left(\frac{1}{2}\right) = 6\left(\frac{1}{2}\right)^3 + 11\left(\frac{1}{2}\right)^2 - \frac{1}{2} - 6$$
3. $$= 6\left(\frac{1}{8}\right) + 11\left(\frac{1}{4}\right) - \frac{1}{2} -6$$
4. $$= \frac{3}{4} + \frac{11}{4} - \frac{2}{4} - 6 = \frac{12}{4} -6 = 3 -6 = -3$$
5. Final remainder = $-3$

*Calculator:* allowed

## Factor Theorem & Cubic Factorisation

**Factor Theorem** — If the remainder when $f(x)$ is divided by $(ax - b)$ is 0, then $(ax - b)$ is a factor of $f(x)$. In other words, if $f\left(\frac{b}{a}\right) = 0$, $(ax - b)$ is a factor.

To factorise a cubic polynomial: first test small integer values to find a root, use the Factor Theorem to get a linear factor, then use coefficient matching or division to find the quadratic factor, then factorise the quadratic if possible.

**Worked example:** Factorise fully $f(x) = x^3 + 3x^2 - 4$.

1. Test small integer values: $f(1) = 1 + 3 -4 = 0$, so $(x -1)$ is a factor
2. Let $x^3 + 3x^2 -4 = (x-1)(Ax^2 + Bx + C)$
3. Expand RHS and equate coefficients: $A=1$, $B=4$, $C=4$
4. Quadratic factor is $x^2 + 4x + 4$, which is a perfect square: $(x+2)^2$
5. Fully factorised form: $(x -1)(x + 2)^2$

> **Exam tip:** Only test integer values of $x$ that divide the cubic's constant term first, as this reduces the number of tests needed.

*Calculator:* allowed

## Combined Exam-Style Problems

Most P2 exam questions combine algebraic division, Remainder Theorem and Factor Theorem into multi-part 4-6 mark questions, often requiring you to solve for unknown coefficients first.

**Worked example:** Given $f(x) = 2x^3 + ax^2 + bx - 6$. When $f(x)$ is divided by $(x-1)$ the remainder is 0; when divided by $(x+2)$ the remainder is 6. (a) Find $a$ and $b$. (b) Factorise $f(x)$ fully.

1. Part (a): Remainder Theorem for $(x-1)$: $f(1) = 2 + a + b - 6 = 0 \implies a + b = 4$.
2. Remainder Theorem for $(x+2)$: $f(-2) = -16 + 4a - 2b - 6 = 6 \implies 4a - 2b = 28 \implies 2a - b = 14$.
3. Solve simultaneously ($a+b=4$, $2a-b=14$): adding gives $3a=18$, so $a=6$, $b=-2$.
4. Part (b): $f(x) = 2x^3 + 6x^2 - 2x - 6 = 2(x^3 + 3x^2 - x - 3)$.
5. Factor by grouping: $x^3+3x^2-x-3 = x^2(x+3) - (x+3) = (x+3)(x^2-1) = (x+3)(x-1)(x+1)$.
6. Final factorised form: $f(x) = 2(x-1)(x+1)(x+3)$.

**Check your understanding**

Independent practice: For $f(x) = 3x^3 -5x^2 -4x +4$, find the remainder when divided by $(3x +1)$, then fully factorise $f(x)$. Show all working.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Using $x = \frac{b}{a}$ for divisor $(ax + b)$ instead of $x = -\frac{b}{a}$
  - Why it fails: Incorrect sign substitution leads to wrong remainders or missed linear factors
  - Correct: Rewrite $(ax + b)$ as $(ax - (-b))$ first, so use $x = -\frac{b}{a}$ for substitution
- **Wrong:** Failing to label quotient and remainder separately when asked
  - Why it fails: Exam mark schemes allocate separate marks for each term, so unlabeled answers lose points
  - Correct: Explicitly write 'Quotient = [expression]' and 'Remainder = [value]' in your answer
- **Wrong:** Testing random integer values for the Factor Theorem
  - Why it fails: Wastes exam time and increases risk of arithmetic error
  - Correct: Only test integer values of $x$ that divide the constant term of the cubic (e.g. for constant -6, test ±1, ±2, ±3, ±6)
- **Wrong:** Leaving a factorisable quadratic factor unfactorised
  - Why it fails: Questions ask for full factorisation, so partial answers lose marks
  - Correct: Always check if the quadratic factor can be split into linear terms or is a perfect square
- **Wrong:** Using full long division when coefficient matching is faster
  - Why it fails: Long division has higher risk of sign errors for linear divisors
  - Correct: Use coefficient matching for linear divisors to save time and reduce error

## Cheatsheet

| Concept | Rule / Formula | Exam Use Case |
| --- | --- | --- |
| Algebraic Division (linear divisor) | $f(x) = (ax\pm b)Q(x) + R$, $\deg(Q) = \deg(f)-1$, $R$ constant | Find quotient and remainder of polynomial division |
| Remainder Theorem | Remainder = $f\left(\frac{b}{a}\right)$ for divisor $(ax - b)$ | Calculate remainder without full algebraic division |
| Factor Theorem | If $f\left(\frac{b}{a}\right) = 0$, $(ax - b)$ is a factor of $f(x)$ | Find linear factors of cubic polynomials |
| Cubic Factorisation Steps | 1. Test root → 2. Get linear factor → 3. Find quadratic factor → 4. Factorise quadratic | Fully factorise cubic expressions |

## What's next

Now that you have mastered algebraic division, Factor and Remainder Theorems for Edexcel IAL P2, you are ready to progress to other core P2 topics. These foundational algebra skills are required for coordinate geometry, differentiation, and trigonometry topics later in P2, as well as for advanced polynomial work in P3 and P4. These concepts are tested in almost every P2 paper, often combined with other topics in multi-part questions, so practice regularly to build speed and accuracy. Always show full working for all steps, even if you use a calculator to check your answers, to ensure you earn all available method marks in your exam.

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