Year 10: Factorising polynomials

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If expanding brackets is like unpacking a parcel, factorising is repacking it neatly. You rewrite a polynomial as a product of simpler factors. In Year 10, that usually means turning a messy sum into brackets you can later cancel, solve, or graph.

Work through the sections in order, then try the linked Year 10 skill quizzes after each section.

1. Why factorise?

• Solving equations:$x\mathrm{²}–5x+6=0$ becomes $(x–2)(x–3)=0$, giving roots quickly.
• Simplifying algebraic fractions: Cancelling common factors prevents heavy algebra later.
• Finding intercepts: In coordinate geometry, factors reveal where a curve crosses the axes.

2. Highest common factor (HCF)

Always start by looking for a common factor in every term.

$6x\mathrm{³}–9x\mathrm{²}=3x\mathrm{²}(2x–3)$

Exercise

3. Factorising by grouping

Useful when four terms appear.

$ax+ay+bx+by=a(x+y)+b(x+y)=(x+y)(a+b)$

Exercise

4. Quadratic trinomials (monic)

For $x\mathrm{²}+bx+c$, find two numbers that multiply to c and add to b.

$x\mathrm{²}+7x+10\text{Numbers}:5\text{and}2\to (x+5)(x+2)$

5. Quadratics where the leading coefficient ≠ 1

Use the “split‑the‑middle” or “cross” method.

$6x\mathrm{²}+11x+4\text{Product}ac=24,\text{factors}8\&36x\mathrm{²}+8x+3x+4=2x(3x+4)+1(3x+4)=(3x+4)(2x+1)$

6. Special products (work backwards)

• Difference of squares:$a\mathrm{²}–b\mathrm{²}=(a+b)(a–b)$
• Perfect square trinomials:$a\mathrm{²}\pm 2ab+b\mathrm{²}=(a\pm b)\mathrm{²}$

$9y\mathrm{²}–16=(3y+4)(3y–4)x\mathrm{²}+10x+25=(x+5)\mathrm{²}$

Exercise

7. Check by expanding

Multiply your factors to confirm they reproduce the original polynomial. A quick FOIL saves exam marks.

8. Common mistakes

• Forgetting the HCF: Always pull it out first; it makes later steps easier.
• Sign errors: Keep track of negatives when splitting the middle term.
• Assuming every quadratic factorises nicely: Some require the quadratic formula. If no integer pair fits, move on.
• Ignoring special products: Recognising a difference of squares saves time and reduces errors.

9. Where you’ll use factorisation

From projectile motion in physics to profit‑loss models in commerce, factorisation helps locate turning points, intercepts, and optimal values. Nail it now and senior maths topics like calculus and polynomial division will feel far less daunting.

So next time you face a polynomial, scan for a common factor, look for patterns, split the middle term if needed, and always expand to check. Factorising is just reverse expansion—master the steps and brackets will bow to your command!