The #1 Rule: You Can Only Cancel FACTORS#
This single rule causes more lost marks than almost anything else in matric:
You can cancel a factor (something multiplied). You can NEVER cancel a term (something added or subtracted).
What this means#
$$ \frac{2x}{x} = 2 \quad \checkmark \quad (x \text{ is a factor of the numerator}) $$$$ \frac{x + 3}{x} \neq 3 \quad \times \quad (x \text{ is a term, not a factor of } x + 3) $$$$ \frac{(x+3)(x-2)}{(x+3)} = x - 2 \quad \checkmark \quad ((x+3) \text{ is a factor of the whole numerator}) $$The “Invisible 1” Test#
If you cancel something from the numerator and the numerator disappears, there must be a 1 left — not zero.
$$ \frac{x}{x(x+1)} = \frac{1}{x+1} \quad \checkmark $$NOT $\frac{0}{x+1}$. When you cancel $x$, a 1 remains.
1. The Factoring Toolkit#
These are the factoring techniques you MUST have memorised for Grade 12:
Common Factor#
Always check this FIRST. Take out the biggest thing common to every term.
$6x^3 - 9x^2 + 3x = 3x(2x^2 - 3x + 1)$
Difference of Two Squares#
$$ a^2 - b^2 = (a - b)(a + b) $$Examples:
- $x^2 - 9 = (x-3)(x+3)$
- $4x^2 - 25 = (2x-5)(2x+5)$
- $\cos^2\theta - \sin^2\theta = (\cos\theta - \sin\theta)(\cos\theta + \sin\theta)$
This last one appears constantly in trig identity proofs!
Trinomials ($ax^2 + bx + c$)#
Find two numbers that multiply to give $ac$ and add to give $b$.
$x^2 + 5x + 6 = (x+2)(x+3)$ — because $2 \times 3 = 6$ and $2 + 3 = 5$.
$2x^2 - 7x + 3 = (2x - 1)(x - 3)$
Sum and Difference of Cubes#
$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$These appear in polynomials and when solving cubic equations.
Grouping (4 terms)#
When you have 4 terms, group them in pairs and factor each pair:
$x^3 + x^2 - 4x - 4 = x^2(x + 1) - 4(x + 1) = (x + 1)(x^2 - 4) = (x+1)(x-2)(x+2)$
2. When to Factorise in Grade 12#
| Situation | Why you need to factor |
|---|---|
| Setting $f'(x) = 0$ | To find the $x$-values of turning points |
| Proving trig identities | To simplify or match the other side |
| Solving equations | To find roots by setting each factor = 0 |
| Simplifying before differentiation | To cancel and reduce to $ax^n$ form |
| Finding x-intercepts of cubics | Factor theorem → divide → factor the quadratic |
3. Cancelling in Practice#
Example 1: Correct Cancelling#
$\frac{x^2 - 4}{x + 2} = \frac{(x-2)(x+2)}{(x+2)} = x - 2$
Factor first, THEN cancel the common factor.
Example 2: WRONG Cancelling#
$\frac{x^2 + 4}{x + 2}$ — this CANNOT be simplified by cancelling!
$x^2 + 4$ does not factorise (it’s a sum of squares, not a difference). There is no $(x+2)$ factor in the numerator.
Example 3: Cancelling with Trig#
$\frac{\sin^2\theta + \sin\theta\cos\theta}{\sin\theta} = \frac{\sin\theta(\sin\theta + \cos\theta)}{\sin\theta} = \sin\theta + \cos\theta$
Factor the numerator first, then cancel.
Example 4: The “$\div x$” Trap in Calculus#
When preparing $\frac{x^3 + 2x}{x^2}$ for differentiation, you split (not cancel):
$= \frac{x^3}{x^2} + \frac{2x}{x^2} = x + 2x^{-1}$
Each term is divided separately. This is correct because you’re dividing each term by the same single-term denominator.
4. Factoring Trig Expressions#
Trig expressions follow the SAME factoring rules as algebra:
| Algebraic | Trigonometric equivalent |
|---|---|
| $a^2 - b^2 = (a-b)(a+b)$ | $\cos^2\theta - \sin^2\theta = (\cos\theta - \sin\theta)(\cos\theta + \sin\theta)$ |
| $a^2 + 2ab + b^2 = (a+b)^2$ | $\sin^2\theta + 2\sin\theta\cos\theta + \cos^2\theta = (\sin\theta + \cos\theta)^2$ |
| $2a^2 - a = a(2a - 1)$ | $2\sin^2\theta - \sin\theta = \sin\theta(2\sin\theta - 1)$ |
🚨 Common Mistakes#
- Cancelling terms: $\frac{x + 5}{x} \neq 5$. This is the single biggest algebraic error. ALWAYS factor first.
- Not factoring completely: $x^2 - 4$ is not fully factored. It becomes $(x-2)(x+2)$.
- Forgetting the common factor: Before using any fancy technique, always look for a common factor first. $2x^2 - 8 = 2(x^2 - 4) = 2(x-2)(x+2)$.
- Sign errors in grouping: When factoring by grouping, if the second group starts with a negative term, factor out $-1$: $x^2 - 3x - 2x + 6 = x(x-3) - 2(x-3) = (x-3)(x-2)$.
- Treating $a^2 + b^2$ as factorisable: $x^2 + 4$ does NOT factor over the real numbers. Only $a^2 - b^2$ factors.
Build on Your Lower-Grade Foundations#
If this still feels shaky, revise the source lessons where these skills are first built:
- Grade 10 Fundamentals: Basic Algebra
- Grade 10 Algebra: Factorization
- Grade 11 Fundamentals: Factorisation Toolkit
- Grade 11 Equations: Quadratic Equations
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