Edexcel IAL P2 Algebraic Methods: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P2 algebraic methods means dividing a polynomial by a linear factor, then using the factor and remainder theorems: $(x - a)$ is a factor of $f (x)$ exactly when $f (a) = 0$ , and the remainder on dividing by $(x - a)$ is $f (a)$ . Use them to factorise cubics fully and to find unknown coefficients.

P2 · Pure Mathematics 2
Calculator allowed
3 worked examples
30 practice questions
Tested in 4 of the last 4 papers

Practise P2 algebraic division and the factor theorem →

Notes
Formulas
Worked examples
Mistakes
FAQ

P2 Algebraic Methods notes

Algebraic division

Divide a polynomial by $(x + a)$ or $(a x + b)$ by long division or by comparing coefficients. Write a zero for any missing power. For example $x^{3} + 2 x^{2} - 5 x - 6 = (x + 1) (x^{2} + x - 6)$ , and the quadratic factorises, so $x^{3} + 2 x^{2} - 5 x - 6 = (x + 1) (x + 3) (x - 2)$ .

The factor theorem

$(x - a)$ is a factor of $f (x)$ if and only if $f (a) = 0$ ; $(a x - b)$ is a factor if and only if $f (\frac{b}{a}) = 0$ . To factorise $f (x) = 2 x^{3} - 3 x^{2} - 11 x + 6$ , try small values: $f (3) = 0$ , so $(x - 3)$ is a factor, and dividing gives $f (x) = (x - 3) (2 x - 1) (x + 2)$ .

The remainder theorem

When $f (x)$ is divided by $(x - a)$ the remainder is $f (a)$ ; dividing by $(a x - b)$ leaves $f (\frac{b}{a})$ . For $f (x) = x^{3} - 4 x + 5$ , the remainder on dividing by $(x - 2)$ is $5$ , and on dividing by $(2 x - 1)$ it is $f (\frac{1}{2}) = \frac{25}{8}$ . Two facts like these give two equations when $f (x)$ has two unknown coefficients.

Where proof fits

The same P2 unit also asks for proof by deduction, by exhaustion and disproof by counter-example. That has its own page: IAL P2 proof.

Key formulas

Factor & remainder theorems Learn it: $f (a) = 0 ⟺ (x - a) a factor; remainder f (a)$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P2 formulas on one page (free PDF) →

P2 Algebraic Methods worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

Let $f (x) = x^{3} + c x^{2} + d x - 4$ . Given $f (x)$ has a factor $(x + 2)$ and leaves a remainder of $- 10$ when divided by $(x - 3)$ , find the values of $c$ and $d$ .

Mark scheme

M1 for applying the Factor Theorem for $(x + 2)$ : $f (- 2) = 0 ⟹ - 8 + 4 c - 2 d - 4 = 0 ⟹ 4 c - 2 d = 12$ .
M1 for applying the Remainder Theorem for $(x - 3)$ : $f (3) = - 10 ⟹ 27 + 9 c + 3 d - 4 = - 10 ⟹ 9 c + 3 d = - 33$ .
M1 for simplifying both equations (e.g., $2 c - d = 6$ and $3 c + d = - 11$ ).
M1 for solving the simultaneous equations (e.g., adding them gives $5 c = - 5$ ).
A1 for finding the correct values $c = - 1$ and $d = - 8$ .

5 steps · 1 to fill in

Worked example 2 · 5 marks · hard

When $f (x) = 2 x^{3} + p x^{2} + q x + 1$ is divided by $(2 x - 1)$ the remainder is 2. When $f (x)$ is divided by $(x + 1)$ the remainder is -4. Find the values of $p$ and $q$ .

Mark scheme

M1 for applying the Remainder Theorem with $x = 0.5$ : $f (0.5) = 2 ⟹ 2 (\frac{1}{8}) + \frac{p}{4} + \frac{q}{2} + 1 = 2$ .
M1 for simplifying the first equation: $\frac{1}{4} + \frac{p}{4} + \frac{2 q}{4} = 1 ⟹ p + 2 q = 3$ .
M1 for applying the Remainder Theorem with $x = - 1$ : $f (- 1) = - 4 ⟹ - 2 + p - q + 1 = - 4 ⟹ p - q = - 3$ .
M1 for solving the simultaneous equations (e.g., subtracting them gives $3 q = 6$ ).
A1 for stating $q = 2$ and $p = - 1$ .

5 steps · 1 to fill in

Worked example 3 · 4 marks · medium

Fully factorise the polynomial $f (x) = 2 x^{3} - 5 x^{2} - 4 x + 3$ .

Mark scheme

M1 for using the factor theorem to find one root, e.g., evaluating $f (- 1) = - 2 - 5 + 4 + 3 = 0$ , to deduce $(x + 1)$ is a factor.
M1 for performing algebraic division to find the quadratic quotient.
A1 for obtaining the correct quadratic quotient $2 x^{2} - 7 x + 3$ .
A1 for fully factorising the quadratic to provide the final answer $(x + 1) (2 x - 1) (x - 3)$ .

4 steps

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Common P2 algebraic division and the factor theorem mistakes

Dropping a missing term. Dividing $x^{3} - 7 x + 6$ ? Write it as $x^{3} + 0 x^{2} - 7 x + 6$ so the columns line up.
The wrong sign of a. $(x + 2)$ is a factor when $f (- 2) = 0$ , not $f (2) = 0$ .
Dividing by (2x − 1) with x = 2. The remainder on dividing by $(2 x - 1)$ is $f (\frac{1}{2})$ : solve $2 x - 1 = 0$ for the value to substitute.
Stopping at one factor. "Factorise fully" means the cubic as a product of three linear factors where possible: factorise the quadratic quotient too.

Every P2 mistake from the examiner reports, with the paper it came from →

P2 Algebraic Methods FAQ

What is the difference between the factor theorem and the remainder theorem?

They are the same idea. The remainder theorem says the remainder on dividing f(x) by (x − a) is f(a); the factor theorem is the special case where that remainder is zero, so (x − a) is a factor.

How do I find which value to try first?

For a cubic with whole-number coefficients, try factors of the constant term divided by factors of the leading coefficient: ±1, ±2, ±3 and so on. One zero gives you a factor, and division does the rest.

Is a calculator allowed in IAL P2?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P2 included. Method marks are for the working you write down, so show it.

How long is the IAL P2 exam?

Each IAL maths unit, P2 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does algebraic division and the factor theorem come up in P2?

In our mapping of the 4 Pearson P2 papers from June 2024 to June 2025, algebraic division and the factor theorem was tested in 4 of them (8 questions in all). Any topic can be examined in any series.

Next steps

More P2: Sequences and Series · Logarithms and Exponentials · Circles · Further Trigonometry · Proof · The Binomial Expansion · Differentiation (stationary points) · Integration (definite integrals and area)

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Other units: IAL P1 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P2 Algebraic Methods: exam questions and answers, notes and worked examples

Try it first

P2 Algebraic Methods notes

Algebraic division

The factor theorem

The remainder theorem

Where proof fits

Key formulas

P2 Algebraic Methods worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · medium

Mark scheme

Common P2 algebraic division and the factor theorem mistakes

P2 Algebraic Methods FAQ

What is the difference between the factor theorem and the remainder theorem?

How do I find which value to try first?

Is a calculator allowed in IAL P2?

How long is the IAL P2 exam?

How often does algebraic division and the factor theorem come up in P2?

Next steps

Your P2 revision kit