6.2 Factorising Expressions

Expanding removes brackets; factorising puts them back. Because the two skills are inverses, ICAS often tests them together: a question shows an expression and asks which of four factorised (or expanded) forms is equivalent. Papers G-H (Years 9-10) name 'factorise linear expressions' as an explicit skill, and it also appears as a step inside equation solving and area problems. The good news: without a calculator, ICAS numbers stay small, so a systematic routine handles everything.

Factorising as the Reverse of Expanding

When you expand 3(2x + 3) you get 6x + 9. Factorising 6x + 9 asks: what was taken OUT to make this? You look for a common factor of every term and write it outside the brackets:

6x + 9 = 3(2x + 3)

Think of it as 'un-distributing'. If expanding multiplies everything inside by the outside factor, factorising divides every term by that factor.

Finding the Highest Common Factor (HCF)

The factor outside must be the highest common factor - the largest number (and any shared variables) that divides every term.

Worked example: factorise 8x + 12.

• Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12.
• The HCF is 4 (not 2 - that works, but it leaves a common factor behind).
• 8x / 4 = 2x and 12 / 4 = 3, so 8x + 12 = 4(2x + 3).

If you only take out 2 and write 2(4x + 6), the expression is PARTLY factorised. ICAS multiple-choice options usually include the fully factorised form, and 'fully factorised' means the terms inside the brackets share no factor other than 1.

Taking out variables too

When every term contains the same variable, it comes out as well: 5x^2 + 15x = 5x(x + 3). Check: 5x x x = 5x^2 and 5x x 3 = 15x. For the powers, take the LOWER power present in every term: from x^3 and x^2 you take out x^2.

Worked example with numbers and variables together: factorise 12ab - 8a. The HCF of 12 and 8 is 4, and both terms contain a, so take out 4a: 12ab - 8a = 4a(3b - 2). Re-expand: 4a x 3b = 12ab and 4a x (-2) = -8a. Correct. Notice the factor can itself contain a variable - this is the step many students stop short of, writing 4(3ab - 2a) and leaving a common a inside the brackets.

Factorising with Negative Terms

Signs need care in both directions:

• 6x - 18 = 6(x - 3). The minus stays inside because -18 / 6 = -3.
• -4x - 12 = -4(x + 3). Taking out a NEGATIVE factor flips both signs inside. Re-expand to confirm: -4 x x = -4x and -4 x 3 = -12. Correct.
• 10 - 5y = 5(2 - y). It is fine for the constant to come first inside the bracket.

Trap: -3x + 9 factorises to -3(x - 3), not -3(x + 3). Re-expanding -3(x + 3) gives -3x - 9, which has the wrong sign on the constant.

Extension: Simple Quadratics (Senior Papers)

Papers I-J extend to quadratics of the form x^2 + bx + c. The method: find two numbers that MULTIPLY to c and ADD to b.

Worked example: factorise x^2 + 7x + 12.

• Pairs multiplying to 12: (1, 12), (2, 6), (3, 4).
• Only 3 and 4 add to 7.
• So x^2 + 7x + 12 = (x + 3)(x + 4).

With a negative constant, the two numbers have opposite signs: x^2 + 2x - 15 needs numbers multiplying to -15 and adding to +2, which are +5 and -3, giving (x + 5)(x - 3).

Checking by Re-expanding

Every factorisation can be checked in seconds by expanding your answer again - you must land exactly on the original expression. Make this a habit in the exam, because it converts a guess into a certainty at the cost of ten seconds. For 'which expression is equivalent?' items, an alternative check is substitution: choose x = 2 (say) and evaluate the original and your chosen option. If the originals differ from an option's value, that option is eliminated immediately.

How ICAS Phrases These Questions

Typical stems include:

• 'Which expression is equivalent to 12a + 8?' (options like 4(3a + 2), 2(6a + 4), 6(2a + 2), 12(a + 8) - note that 2(6a + 4) is mathematically equal but not FULLY factorised, so read the stem carefully: 'equivalent' accepts any equal form, 'fully factorised' demands the HCF)
• 'The area of a rectangle is 6x + 15 square units. If one side is 3 units, the other side is...' - factorise: 3(2x + 5), so the other side is 2x + 5
• 'Factorise 9y - 27' - straight skill recall: 9(y - 3)

A summary of the routine:

1. Find the HCF of the numbers.
2. Take out any variable present in EVERY term (lowest power).
3. Divide each term by the factor to fill the brackets, watching signs.
4. Re-expand to check; confirm nothing inside still shares a factor.