What is factorisation in algebra and why is it important in the IB MYP Mathematics curriculum?

Factorisation in algebra involves breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. It is crucial in the IB MYP Mathematics curriculum because it helps students simplify expressions, solve equations, and understand the properties of numbers and algebraic expressions more deeply.

How do you factorise a quadratic expression in the form ax^2 + bx + c?

To factorise a quadratic expression like ax^2 + bx + c, you need to find two numbers that multiply to ac (the product of the coefficient of x^2 and the constant term) and add up to b (the coefficient of x). Once these numbers are identified, you can rewrite the middle term and factor by grouping. This method is commonly taught in the IB MYP Mathematics curriculum.

What are common factorisation techniques used in algebra?

Common factorisation techniques include factoring out the greatest common factor (GCF), using the difference of squares, factoring trinomials, and applying the sum or difference of cubes. These techniques are essential for simplifying expressions and solving equations, as emphasized in the IB MYP Mathematics program.

Can you explain the difference between factorisation and expanding in algebra?

Factorisation is the process of breaking down an expression into a product of simpler factors, while expanding involves multiplying out factors to form a single expression. In the IB MYP Mathematics curriculum, students learn both processes to manipulate and simplify algebraic expressions effectively.

Why is understanding factorisation important for solving algebraic equations?

Understanding factorisation is important for solving algebraic equations because it allows students to simplify complex expressions and find solutions more easily. By rewriting equations in a factored form, students can apply the zero-product property to find the roots of equations, a key skill in the IB MYP Mathematics curriculum.

Why is "Forgetting to take out the HCF first." a common mistake in Factorisation?

Why it happens: Diving straight to trinomial factorisation. How to avoid it: ALWAYS check for HCF first. 2x^2 + 10x + 12 has HCF 2: factor out to get 2(x^2 + 5x + 6) = 2(x+2)(x+3). Missing the HCF leaves the problem half-done.

Why is "Trying to factorise $a^2 + b^2$." a common mistake in Factorisation?

Why it happens: Over-applying the difference-of-squares pattern. How to avoid it: a^2 + b^2 has NO factorisation over the real numbers. Only a^2 - b^2 (with a MINUS) factorises.

Why is "Stopping without checking by expansion." a common mistake in Factorisation?

Why it happens: Trusting the answer. How to avoid it: Always expand your factorisation to verify. If it doesn't reproduce the original, you've made an error.

AlgebraIB MYP Maths Extended Level

Factorisation

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Factorisation — IB MYP Mathematics (Extended): the reverse of expansion

Factorisation puts brackets BACK into an expanded expression. This MYP Mathematics Extended note covers HCF factorisation, the difference of two squares, and trinomial factorisation.

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Factorise by taking out the HCF.
MYP Mathematics A — Recognise and factorise difference of two squares.
MYP Mathematics A — Factorise quadratic trinomials.
MYP Mathematics C — Show clear working at each factorisation step.

HCF factorisation — always start here

Take out everything common to all terms.

Step 1 of any factorisation: look for a HCF that divides every term — including any common letter factors. Factor it out front.

Worked example. Factorise $6x^2 + 9x$ .

HCF of numbers: $\text{HCF}(6, 9) = 3$ .
Common letter factor: $x$ (in both terms).
Combined HCF: $3x$ .
Factor out: $6x^2 + 9x = 3x(2x + 3)$ .

Check by expansion: $3x \times 2x + 3x \times 3 = 6x^2 + 9x$ . ✓

Worked example. Factorise $-12a^3b + 8a^2b^2 - 4ab^3$ .

HCF of coefficients: $\text{HCF}(12, 8, 4) = 4$ .
Common letters: $a^1$ (lowest power of $a$ ), $b^1$ (lowest power of $b$ ). So $ab$ .
Combined HCF: $4ab$ .
I'll factor out $-4ab$ to make the leading term inside positive: $-12a^3b + 8a^2b^2 - 4ab^3 = -4ab(3a^2 - 2ab + b^2)$ .

Rule: for variables, take the LOWEST POWER appearing in any term.

Always look for HCF first.
Include common letter factors (lowest power).
Check by expanding the result.

Difference of two squares

$a^2 - b^2 = (a-b)(a+b)$ .

Recognise this pattern: TWO squared terms separated by a MINUS sign:

$a^2 - b^2 = (a - b)(a + b).$

Worked examples:

$x^2 - 9 = x^2 - 3^2 = (x-3)(x+3)$ .
$16y^2 - 25 = (4y)^2 - 5^2 = (4y - 5)(4y + 5)$ .
$49 - p^2 = 7^2 - p^2 = (7 - p)(7 + p)$ .

Combined with HCF:

$2x^2 - 18 = 2(x^2 - 9) = 2(x-3)(x+3)$ . (HCF FIRST, then DOTS.)
$5a^2 - 5b^2 = 5(a^2 - b^2) = 5(a-b)(a+b)$ .

This identity is so useful it's worth memorising in its forward AND backward direction:

Expansion: $(a-b)(a+b) \to a^2 - b^2$ .
Factorisation: $a^2 - b^2 \to (a-b)(a+b)$ .

A common trick: this pattern works for the DIFFERENCE only — $a^2 + b^2$ does NOT factorise over the real numbers.

Geometrically, the difference of two squares cuts and rearranges into a rectangle with sides $(a-b)$ and $(a+b)$.

$a^2 - b^2 = (a-b)(a+b)$ — memorise it.
Spot SQUARES on both sides and a MINUS sign.
$a^2 + b^2$ does NOT factorise over the reals.

Quadratic trinomials

Find two numbers that multiply to $c$ and add to $b$ .

Case 1: leading coefficient 1. Factorise $x^2 + bx + c$ as $(x + p)(x + q)$ where $pq = c$ and $p + q = b$ .

Worked example. Factorise $x^2 + 7x + 12$ .

Need two numbers that multiply to 12 and add to 7.
$3 \times 4 = 12$ and $3 + 4 = 7$ . ✓
$x^2 + 7x + 12 = (x + 3)(x + 4)$ .

Worked example. Factorise $x^2 - 5x + 6$ .

Multiply to 6, add to $-5$ .
Both factors must be NEGATIVE (so product positive, sum negative): $-2 \times -3 = 6$ , $-2 + -3 = -5$ . ✓
$x^2 - 5x + 6 = (x - 2)(x - 3)$ .

Worked example. Factorise $x^2 - x - 12$ .

Multiply to $-12$ , add to $-1$ .
One positive, one negative. $-4 \times 3 = -12$ and $-4 + 3 = -1$ . ✓
$x^2 - x - 12 = (x - 4)(x + 3)$ .

Case 2: leading coefficient $\ne 1$ (Extended). Factorise $ax^2 + bx + c$ .

Use the AC method:

Multiply $a \times c$ .
Find two numbers that multiply to $ac$ and add to $b$ .
Split the middle term using those numbers.
Factor by grouping.

Worked example. Factorise $2x^2 + 7x + 3$ .

$a \times c = 2 \times 3 = 6$ .
Two numbers multiplying to 6 and adding to 7: $6 \times 1 = 6$ , $6 + 1 = 7$ . ✓
Split: $2x^2 + 6x + x + 3$ .
Group: $2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)$ .

So $2x^2 + 7x + 3 = (x + 3)(2x + 1)$ .

Always check by expanding!

Trinomial $x^2 + bx + c$ : find $p, q$ with $pq = c$ and $p + q = b$ .
$ax^2 + bx + c$ (Extended): AC method — multiply, split, group.
Always verify by expanding the brackets.

How it’s examined

Criterion A: factorise expressions of increasing complexity. Criterion C: clear step-by-step working.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

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Step-by-step worked examples — Factorisation

Step-by-step solutions to past-paper-style questions on factorisation, written exactly the way a tutor would explain them at the board.

1HCF factorisation
Getting started• HCF
Question
Factorise $15x^2 - 10x$ .
Step-by-step solution
1. Step 1
  HCF of coefficients: $\text{HCF}(15, 10) = 5$ .
2. Step 2
  Common letter: $x$ (lowest power in both).
3. Step 3
  Factor out $5x$ : $15x^2 - 10x = 5x(3x - 2)$ .
4. Step 4
  Check by expanding: $5x \times 3x = 15x^2$ , $5x \times -2 = -10x$ . ✓
Answer
$5x(3x - 2)$
2Difference of two squares
Building confidence• DOTS
Question
Factorise $25x^2 - 49$ .
Step-by-step solution
1. Step 1
  Recognise as $a^2 - b^2$ : $25x^2 = (5x)^2$ and $49 = 7^2$ .
2. Step 2
  Apply identity: $25x^2 - 49 = (5x - 7)(5x + 7)$ .
Answer
$(5x - 7)(5x + 7)$
3Factorise a simple trinomial
Building confidence• trinomial
Question
Factorise $x^2 - 7x + 12$ .
Step-by-step solution
1. Step 1
  Find two numbers multiplying to $12$ and adding to $-7$ .
2. Step 2
  Both negative (product positive, sum negative): $-3 \times -4 = 12$ and $-3 + -4 = -7$ . ✓
3. Step 3
  $x^2 - 7x + 12 = (x - 3)(x - 4)$ .
Answer
$(x - 3)(x - 4)$
4Factorise with leading coefficient
Stretch• AC method
Question
Factorise $3x^2 + 11x + 6$ .
Step-by-step solution
1. Step 1
  $a \times c = 3 \times 6 = 18$ . Need two numbers multiplying to 18 and adding to 11: $9 + 2 = 11$ and $9 \times 2 = 18$ . ✓
2. Step 2
  Split the middle term: $3x^2 + 9x + 2x + 6$ .
3. Step 3
  Group: $3x(x + 3) + 2(x + 3) = (x + 3)(3x + 2)$ .
Answer
$(x + 3)(3x + 2)$

Key Definitions and Keywords — Factorisation

Definitions to memorise and the exact keywords mark schemes credit for factorisation answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Factorise
Examiner keyword
Write an expression as a product of factors (brackets).
Difference of two squares
Examiner keyword
The identity $a^2 - b^2 = (a-b)(a+b)$ .
Quadratic trinomial
An expression of the form $ax^2 + bx + c$ with three terms.

Common Mistakes and Misconceptions — Factorisation

The traps other students keep falling into on factorisation questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Forgetting to take out the HCF first.
Why it happens
Diving straight to trinomial factorisation.
How to avoid it
ALWAYS check for HCF first. $2x^2 + 10x + 12$ has HCF 2: factor out to get $2(x^2 + 5x + 6) = 2(x+2)(x+3)$ . Missing the HCF leaves the problem half-done.
✕Trying to factorise $a^2 + b^2$ .
Why it happens
Over-applying the difference-of-squares pattern.
How to avoid it
$a^2 + b^2$ has NO factorisation over the real numbers. Only $a^2 - b^2$ (with a MINUS) factorises.
✕Stopping without checking by expansion.
Why it happens
Trusting the answer.
How to avoid it
Always expand your factorisation to verify. If it doesn't reproduce the original, you've made an error.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

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Factorisation — frequently asked questions

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Factorisation

Short Study Notes in the form of Slides

Short Study Notes — Factorisation

Detailed Notes

What you’ll learn

How it’s examined

1HCF factorisation

2Difference of two squares

3Factorise a simple trinomial

4Factorise with leading coefficient

Factorise

Difference of two squares

Quadratic trinomial

✕Forgetting to take out the HCF first.

✕Trying to factorise a2+b2a^2 + b^2a2+b2.

✕Stopping without checking by expansion.

Practice questions

Past paper style quiz

Assess your understanding

Factorisation — frequently asked questions

✕Trying to factorise $a^2 + b^2$ .