Algebra

Factorisation

Common factor, difference of squares, trinomials, and grouping. Always take out the HCF first.

CAPS

Grade 10 · Mathematics · Algebra

This sits on Mathematics Paper 1.

The paper
Paper 1 is algebra, exponents, equations, number patterns, functions, finance and probability. Copy the formula, then substitute.
Cognitive level
CAPS spreads the marks across knowledge, routine procedures, complex procedures and problem solving. A routine question still wants the line of working. A problem-solving question wants you to choose the method, not only to calculate.
The relation
a² − b² = (a − b)(a + b). Difference of squares
How a marker reads it
“Factorise fully” on KZN Paper 1 means HCF AND the remaining identity — leaving 2(x² − 9) loses a mark.

Each Mathematics paper is 150 marks. A correct substitution can earn a method mark even when the last arithmetic slips, if the formula was the right one.

Diagram

Worked example

Factorise fully: 2x² − 18.

Before any working — hunt first

Given

Numbers and units already in the sentence: 2x² · 18. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Factorise fully: 2x² − 18. Circle that word — that is the missing variable.

Find

We are required to find: Factorise fully: 2x² − 18. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write a² − b² = (a − b)(a + b) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Factorise fully: 2x² − 18.

Simplified approach

Pull 2 out first: 2(x² − 9). Then difference of squares.

Culture note

Sharing 2 crates of 9 oranges at a Durban market — factor first, then split the square.

Formulas

a²−b²=(a − b)(a + b)

Difference of squares

x²+(p+q)x+pq=(x+p)(x+q)

Trinomial

How to hunt the missing letter

Do not start by naming the letters. First hunt the missing one: copy every number you can see (Given). The letter with no number is Missing. The asking word is Find. The formula that still contains that letter is what you Calculate.

How to hunt: (1) Underline every number and its unit — those are Given. (2) Circle the asking word — find, calculate, how far, what is. That word names Find. (3) The letter the asking word names, which has no number yet, is Missing. (4) Write the formula with empty letters. Cross out every given letter. The letter still standing is what you Calculate.

a
first square root
How you spot this in the sentence
When the sentence names the first square root, that letter is a. Copy the number next to it into Given. If the sentence never names it, a is Missing. In a² − b², a is the thing being squared first.
Example:x² − 25 = a² − b² with a = x, b = 5.
b
second square root
How you spot this in the sentence
When the sentence names the second square root, that letter is b. Copy the number next to it into Given. If the sentence never names it, b is Missing. The number whose square is subtracted.
Example:In 9x² − 16, b = 4 because 16 = 4².
p, q
factor pair
How you spot this in the sentence
When the sentence names the factor pair, that letter is p, q. Copy the number next to it into Given. If the sentence never names it, p, q is Missing. Two numbers that add to the middle coefficient and multiply to the last term.
Example:x²+5x+6=(x + 2)(x + 3), so p=2, q=3.

Exam tricks you must know

These are the moves markers look for — not shortcuts that skip the method, but the habits that earn the method mark.

“Factorise fully” means HCF and then the identity — leaving 2(x² − 9) loses a mark
How:Look at the given polynomial. Pull out the biggest number and letter that divides every term (HCF). Then look at what is left: difference of squares, trinomial, or grouping. Both layers go on the page. Stop only when nothing else factorises.
When to use it:Any “factorise fully” on Paper 1, especially KZN.
Memorise exact values for 0°, 30°, 45°, 60°, 90° — compound angles collapse without them
How:Before a calculator, write the triangle: 30-60-90 and 45-45-90. sin 30° = 12, sin 45° = 1√2, sin 60° = √32, and the cosines swap. Compound-angle questions want these as fractions with division lines, not decimals.
When to use it:Reduction, compound angles, and “without a calculator” trig.
Name similar triangles in corresponding order: ΔABC ∼ ΔDEF, not ΔABC ∼ ΔEFD
How:Hunt equal angles first and mark them on the sketch. Then list vertices so that equal angles sit in the same position: A maps to D, B to E, C to F. The ratio statement uses that same order. Wrong order loses the reason mark even if the lengths are right.
When to use it:Similarity, intercept, and 3-D trig that shares a triangle.

Step 1 / 5

HCF

2x²−18=2(x² − 9).

2(x² − 9)

Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.

4. Step-by-step working

01
HCF
2x²−18=2(x² − 9).
2(x² − 9)
02
Squares
x²−9=(x − 3)(x + 3).

More worked examples

Board walkthroughs on this topic. Open one, copy the working, then try the next.

Example 1 / 20

Factorise x² + 5x + 6.

4. Step-by-step working

01
Identify the given
Numbers 2 and 3: 2+3 = 5, 2×3 = 6.
2 and 3
02
Write as a product
(x + 2)(x + 3).
(x + 2)(x + 3)

How we got there

Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.

Explain in another language · 11

1. What is the question asking?

Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Factorise x² + 5x + 6.

Before any working — hunt first

Given

Numbers and units already in the sentence: 5x · 6. Copy each one into a column. A name without a number is not given.

Missing

After you copy the given, the letter that still has no number is missing. The paper is pointing at: Factorise x² + 5x + 6. Circle that word — that is the missing variable.

Find

We are required to find: Factorise x² + 5x + 6. Write that as a symbol on the left, with a question mark, before you touch a formula.

Calculate

Write 2 and 3 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Factorise x² + 5x + 6.

In other words — as if you have never seen this before

If you have never seen this before: Look for p, q with p+q = b and pq = c in x² + bx + c. We will name a method in words, write the formula with no numbers, then substitute one line at a time.

2. What do we already know?

Trinomial x² + 5x + 6. Find two numbers that multiply to the constant and add to the x-coefficient.

3. Which method — and why this one?

Look for p, q with p+q = b and pq = c in x² + bx + c.

4. Step-by-step working

01 / 05
Identify the given
Numbers 2 and 3: 2+3 = 5, 2×3 = 6.
2 and 3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Write as a product
(x + 2)(x + 3).
(x + 2)(x + 3)
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
(x + 2)(x + 3)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.

5. The answer

(x + 2)(x + 3)

Copy onto your answer sheet — in this order

Formula: 2 and 3

Answer: (x + 2)(x + 3)

6. Check it like a marker

Expand (x + 2)(x + 3) to recover x² + 5x + 6.

7. A trap first-timers fall into

First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”

Share on WhatsApp

Mathematics

Take topic quiz