Why Factorization Matters#
Factorization is the reverse of expansion. If expansion is “building” a wall from bricks, factorization is “taking the wall apart” into its original bricks.
You will use factorization in:
- Solving equations (setting each factor = 0)
- Simplifying algebraic fractions (cancelling common factors)
- Trigonometric identities (Grade 11–12)
- Finding x-intercepts of graphs
- Calculus (Grade 12)
The Golden Rule: Always Check for a Common Factor FIRST#
Before using any other technique, always pull out the Highest Common Factor (HCF) of all terms.
$6x^3 + 9x^2 - 3x = 3x(2x^2 + 3x - 1)$
If you skip this step, everything else becomes harder.
1. Common Factor#
Look for numbers and variables that appear in every term.
Worked Examples#
$5x + 10 = 5(x + 2)$
$3x^2 - 12x = 3x(x - 4)$
$8a^2b + 12ab^2 = 4ab(2a + 3b)$
$-2x^2 + 6x = -2x(x - 3)$ ← Factor out the negative too!
2. Difference of Two Squares (DOTS)#
$$ a^2 - b^2 = (a - b)(a + b) $$Requirement: Must be a SUBTRACTION of two perfect squares.
Worked Examples#
$x^2 - 25 = (x - 5)(x + 5)$
$4x^2 - 9 = (2x)^2 - (3)^2 = (2x - 3)(2x + 3)$
$x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)$
← Factor again! $x^2 - 4$ is itself a DOTS. But $x^2 + 4$ cannot be factored further (it’s a SUM of squares).
$3x^2 - 27 = 3(x^2 - 9) = 3(x - 3)(x + 3)$ ← Common factor first, THEN DOTS.
⚠️ $a^2 + b^2$ (a SUM of squares) CANNOT be factored. Only the difference works.
3. Trinomials ($x^2 + bx + c$)#
Find two numbers that multiply to give $c$ and add to give $b$.
The Sign Logic#
| Sign of $c$ | Sign of $b$ | What the two numbers look like |
|---|---|---|
| $c > 0$ | $b > 0$ | Both positive: $(x + ?)(x + ?)$ |
| $c > 0$ | $b < 0$ | Both negative: $(x - ?)(x - ?)$ |
| $c < 0$ | $b > 0$ | Different signs, larger one positive |
| $c < 0$ | $b < 0$ | Different signs, larger one negative |
Worked Examples#
Example 1: $x^2 + 7x + 12$
Need: multiply to 12, add to 7 → 3 and 4
$= (x + 3)(x + 4)$
Example 2: $x^2 - 5x + 6$
Need: multiply to 6, add to $-5$ → $-2$ and $-3$
$= (x - 2)(x - 3)$
Example 3: $x^2 + 2x - 15$
Need: multiply to $-15$, add to 2 → 5 and $-3$
$= (x + 5)(x - 3)$
Example 4: $x^2 - x - 20$
Need: multiply to $-20$, add to $-1$ → $-5$ and 4
$= (x - 5)(x + 4)$
4. Harder Trinomials ($ax^2 + bx + c$ where $a \neq 1$)#
When the coefficient of $x^2$ is not 1, 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#
$2x^2 + 7x + 3$
Step 1: $a \times c = 2 \times 3 = 6$
Step 2: Numbers that multiply to 6 and add to 7 → 6 and 1
Step 3: Split: $2x^2 + 6x + x + 3$
Step 4: Group: $2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)$
Another Example#
$3x^2 - 10x + 8$
$ac = 24$. Numbers: $-6$ and $-4$ (multiply to 24, add to $-10$)
$= 3x^2 - 6x - 4x + 8 = 3x(x - 2) - 4(x - 2) = (x - 2)(3x - 4)$
5. Grouping (4 Terms)#
When you have 4 terms, group them in pairs and factor each pair.
Worked Example#
$x^3 + x^2 - 9x - 9$
Group: $(x^3 + x^2) + (-9x - 9)$
$= x^2(x + 1) - 9(x + 1)$
$= (x + 1)(x^2 - 9)$
$= (x + 1)(x - 3)(x + 3)$ ← DOTS again!
Another Example#
$2xy + 3x + 2y + 3$
$= x(2y + 3) + 1(2y + 3) = (2y + 3)(x + 1)$
Tip: If the first grouping doesn’t give a common bracket, try rearranging the 4 terms into different pairs.
6. Sum and Difference of Cubes#
$$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$$$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$Memory aid: “Same sign, opposite sign, always positive”:
- First bracket: same sign as the original
- Second bracket: first term positive, middle term opposite, last term always positive
Worked Examples#
$x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)$
$27x^3 + 1 = (3x)^3 + 1^3 = (3x + 1)(9x^2 - 3x + 1)$
7. The Complete Strategy (Flowchart)#
For any expression:
- Common factor? → Take it out.
- How many terms?
- 2 terms: Is it DOTS? Sum/Difference of cubes?
- 3 terms: Is it a trinomial? Use the sign logic or “ac” method.
- 4 terms: Try grouping.
- Can you factor further? → Check each bracket again.
🚨 Common Mistakes#
- Skipping the common factor: $2x^2 + 10x + 12 = 2(x^2 + 5x + 6) = 2(x+2)(x+3)$. If you skip the 2, you get $(2x + 4)(x + 3)$ which is messier and easier to get wrong.
- Not factoring completely: $(x^2 - 4)$ is NOT fully factored. It’s still DOTS: $(x-2)(x+2)$.
- $a^2 + b^2$ is NOT factorable: Only the difference $a^2 - b^2$ factors.
- Sign errors in trinomials: If $c$ is negative, the two numbers have DIFFERENT signs. Don’t make them both positive.
- Grouping with wrong sign: $-9x - 9 = -9(x + 1)$, NOT $9(x - 1)$. Factor out the NEGATIVE.
💡 Pro Tip: The FOIL Check#
After factoring, ALWAYS expand your answer mentally (or on paper) to verify it gives back the original expression. This takes 10 seconds and catches most errors.
🔗 Related Grade 10 topics:
- Multiplying Brackets — expanding is the REVERSE of factoring. Use it to check your answers.
- Solving Equations — you factorise to solve quadratic equations (zero product rule)
- Sketching Graphs — x-intercepts of a parabola come from factoring $ax^2 + q = 0$
📌 Where this leads:
- Grade 11 Equations — factoring is the #1 method for solving quadratic equations
- Grade 12 Fundamentals: Factoring & Cancelling — the factoring toolkit you MUST know for matric
- Grade 12 Polynomials — factor theorem and solving cubic equations