MathsICSE Class 9 · Factorisation

Factorisation by Common Factors, Grouping, Squares and Cubes (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

First take out any common factor, then look for a pattern: a² − b² = (a − b)(a + b), a² ± 2ab + b² = (a ± b)², a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). For four terms try grouping. Keep factorising until no factor can be broken further.

Always take out the HCF first, let the number of terms choose the pattern, then keep factorising and check by multiplying back.

Illustration drawn for QuestionBuddy.

Understand Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes

Order of attack. (1) Take out the highest common factor (HCF). (2) Count the terms: two terms suggest a difference of squares or cubes; three suggest a perfect square or splitting the middle term; four suggest grouping. (3) Check whether each factor can be factorised further.

Common factor and grouping. $a x + a y + b x + b y = a (x + y) + b (x + y) = (a + b) (x + y)$ . For $x^{3} - 3 x^{2} + 2 x - 6$ : group $x^{2} (x - 3) + 2 (x - 3) = (x - 3) (x^{2} + 2)$ . If the grouping you chose does not give a common bracket, rearrange the terms.

Difference of two squares. $a^{2} - b^{2} = (a - b) (a + b)$ . Apply repeatedly: $x^{4} - 81 = (x^{2} - 9) (x^{2} + 9) = (x - 3) (x + 3) (x^{2} + 9)$ . The factor $x^{2} + 9$ is a sum of squares and does not break further into real factors.

Perfect squares together with a square. $4 a^{2} - 12 ab + 9 b^{2} - c^{2} = (2 a - 3 b)^{2} - c^{2} = (2 a - 3 b - c) (2 a - 3 b + c)$ . Look for three terms that form a perfect square, then use a difference of squares.

Sum and difference of cubes.
$ $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}), a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2}) .$ $
The sign inside the second bracket is opposite to the sign in the first for the middle term, and the last term is always $+ b^{2}$ . Example: $27 a^{3} - 8 b^{3} = (3 a)^{3} - (2 b)^{3} = (3 a - 2 b) (9 a^{2} + 6 ab + 4 b^{2})$ .

Using factors to compute. $100 3^{2} - 99 7^{2} = (1003 - 997) (1003 + 997) = 6 \times 2000 = 12000$ . For $8. 7^{3} + 1. 3^{3}$ use $a + b = 10$ : $10 (8. 7^{2} - 8.7 \times 1.3 + 1. 3^{2}) = 10 \times 66.07 = 660.7$ .

Check. Multiply the factors back, or test with $x = 1$ or $x = 2$ .

Level. Class IX is a school-exam class. Items marked advanced chain two methods; the hots items need a clean sum-of-cubes or difference-of-squares trick for a number and are at the top of the school range.

Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes: formulas to remember

Difference of squares: $a^{2} - b^{2} = (a - b) (a + b)$; Two terms, both squares.
Perfect squares: $a^{2} \pm 2 ab + b^{2} = (a \pm b)^{2}$; Three terms.
Sum of cubes: $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$; Last term +b².
Difference of cubes: $a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})$; Middle term +ab.
Grouping: $a x + a y + b x + b y = (a + b) (x + y)$; Four terms.

How to solve Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes questions

Common factor first, then a pattern, then check for further factors.

1Take out the HCF of all terms (including the numerical part).
2Count the terms to choose a method.
3Write each term as a perfect square or cube where relevant to identify $a$ and $b$ .
4Apply the identity and write the factors in brackets.
5Examine each factor for a further difference of squares or cubes.
6Multiply back (or test with a number) to verify.

Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Path around a square lawn

Suppose a square lawn of side 13 m sits inside a square plot of side 17 m, and the strip between them is paved.

Find the paved area using factors.

Try it first, then show the solutionSolution

Area $= 1 7^{2} - 1 3^{2} = (17 - 13) (17 + 13) = 4 \times 30 = 120$ m².

Answer: 120 m².

Memory hook: Difference of squares is the area of the frame.

Arithmetic in the head

Imagine a tile maker comparing two square tiles of sides 1003 mm and 997 mm.

Find the difference of their areas in mm².

Try it first, then show the solutionSolution

$100 3^{2} - 99 7^{2} = (1003 - 997) (1003 + 997) = 6 \times 2000 = 12000$ .

Answer: 12000 mm².

Memory hook: Factorise instead of squaring big numbers.

Tricks to remember Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes

Short memory aids. Say each one aloud once, then use it on a question the same day.

HCF first
Always pull out the common factor before anything else.
Use it when: Avoiding messy identities.
Sum of squares stays
a² + b² does not factorise over real numbers.
Use it when: Stopping correctly.
SOAP for cubes
Same, Opposite, Always Positive for the signs in (a ± b)(a² ∓ ab + b²).
Use it when: Cube factors.

Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Difference of squares · 1 mark · MCQBoard level

$49 x^{2} - 81 y^{2}$ factorises as:
(a) $(7 x - 9 y)^{2}$
(b) $(7 x - 9 y) (7 x + 9 y)$
(c) $(49 x - 81 y) (x + y)$
(d) it cannot be factorised

Try it first, then show the solutionSolution

$(7 x)^{2} - (9 y)^{2} = (7 x - 9 y) (7 x + 9 y)$ .

Answer: (b)

Example 2: Grouping · 3 marks · Short answerBoard level

Factorise $x^{3} - 3 x^{2} + 2 x - 6$ .

Try it first, then show the solutionSolution

Group: $x^{2} (x - 3) + 2 (x - 3)$ .
$= (x - 3) (x^{2} + 2)$ .

Answer: $(x - 3) (x^{2} + 2)$

Example 3: Cubes · 4 marks · Short answerHarder board level

Factorise (i) $27 a^{3} - 8 b^{3}$ (ii) $64 x^{3} + 1$ .

Try it first, then show the solutionSolution

(i) $(3 a)^{3} - (2 b)^{3} = (3 a - 2 b) (9 a^{2} + 6 ab + 4 b^{2})$ .
(ii) $(4 x)^{3} + 1^{3} = (4 x + 1) (16 x^{2} - 4 x + 1)$ .

Answer: (i) $(3 a - 2 b) (9 a^{2} + 6 ab + 4 b^{2})$ (ii) $(4 x + 1) (16 x^{2} - 4 x + 1)$

Example 4: Repeated and combined identities · 4 marks · Short answerHarder board level

Factorise completely (i) $x^{4} - 81$ (ii) $4 a^{2} - 12 ab + 9 b^{2} - c^{2}$ .

Try it first, then show the solutionSolution

(i) $x^{4} - 81 = (x^{2} - 9) (x^{2} + 9) = (x - 3) (x + 3) (x^{2} + 9)$ .
(ii) $(2 a - 3 b)^{2} - c^{2} = (2 a - 3 b - c) (2 a - 3 b + c)$ .

Answer: (i) $(x - 3) (x + 3) (x^{2} + 9)$ (ii) $(2 a - 3 b - c) (2 a - 3 b + c)$

Example 5: Numbers with factors · 5 marks · Long answerHOTS / competency

Without a calculator find (i) $100 3^{2} - 99 7^{2}$ (ii) $8. 7^{3} + 1. 3^{3}$ , giving the working.

Try it first, then show the solutionSolution

(i) $(1003 - 997) (1003 + 997) = 6 \times 2000 = 12000$ .
(ii) $a = 8.7$ , $b = 1.3$ , $a + b = 10$ .
$a^{2} = 75.69$ , $ab = 11.31$ , $b^{2} = 1.69$ , so $a^{2} - ab + b^{2} = 75.69 - 11.31 + 1.69 = 66.07$ .
$a^{3} + b^{3} = 10 \times 66.07 = 660.7$ .
Check: $658.503 + 2.197 = 660.7$ . ✓

Answer: 12000; 660.7

Where students lose marks in Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes

The slips examiners see most often, and what to write instead.

Mistake: Stopping before the factorisation is complete
A factor like x² − 9 still breaks.
Wrong working
$x^{4} - 81 = (x^{2} - 9) (x^{2} + 9)$ .
Right working
$(x - 3) (x + 3) (x^{2} + 9)$ .
Do this: Check each bracket.
Usually costs: Final mark.
Mistake: Wrong sign in a cube factor
The last term is always positive.
Wrong working
$a^{3} - b^{3} = (a - b) (a^{2} - ab - b^{2})$ .
Right working
$(a - b) (a^{2} + ab + b^{2})$ .
Do this: Remember SOAP.
Usually costs: Whole answer.
Mistake: Skipping the common factor
It makes the next step easier.
Wrong working
$2 x^{2} - 18$ : cannot factorise.
Right working
$2 (x^{2} - 9) = 2 (x - 3) (x + 3)$ .
Do this: HCF first.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; "factorise completely" means keep going until nothing breaks.
In a number question, say which identity you are using (for example "difference of two squares").
Write the sums of squares and cubes side by side to avoid sign errors.
Multiply back at least one factor pair to check.

Practice questions on Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Factorise $2 x^{2} - 18$ .
(a) $2 (x - 3)^{2}$
(b) $2 (x - 3) (x + 3)$
(c) $(2 x - 9) (x + 2)$
(d) $2 (x^{2} - 9 x)$
Hint
Take out 2 first.
Answer
(b)
Q2 · 4 marks · Short answerHarder board level
Factorise completely $a^{6} - b^{6}$ .
Hint
Write it as $(a^{3})^{2} - (b^{3})^{2}$ .
Answer
$(a^{3} - b^{3}) (a^{3} + b^{3}) = (a - b) (a^{2} + ab + b^{2}) (a + b) (a^{2} - ab + b^{2})$ .
Q3 · 3 marks · Short answerHarder board level
Factorise $x^{3} - y^{3} - x + y$ .
Hint
Group as $(x^{3} - y^{3}) - (x - y)$ .
Answer
$(x - y) (x^{2} + x y + y^{2}) - (x - y) = (x - y) (x^{2} + x y + y^{2} - 1)$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $x^{2} + 9$ cannot be factorised into real linear factors.
Reason (R): A sum of two squares has no real factorisation of the form $(x - a) (x + a)$ .
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$x^{2} = - 9$ has no real solution.
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A square park of side 17 m has a square pond of side 13 m inside it. The remaining ground is turfed.
(a) Write the turfed area as a difference of squares.
(b) Factorise and evaluate it.
(c) Turf costs ₹50 per m². Find the cost.
Hint
$a^{2} - b^{2}$ with $a = 17$ , $b = 13$ .
Answer
(a) $1 7^{2} - 1 3^{2}$ (b) $(17 - 13) (17 + 13) = 4 \times 30 = 120$ m² (c) $120 \times 50 = ₹6, 000$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Factorise a² − b².Show answer
(a − b)(a + b).
2.Factorise a³ + b³.Show answer
(a + b)(a² − ab + b²).
3.Does x² + 9 factorise?Show answer
No, not into real factors.
4.What is the first step in any factorisation?Show answer
Take out the common factor.

Factorising by Common Factors, Grouping, Difference of Squares and Sum or Difference of Cubes: frequently asked questions

How do I know which method to use?

Count the terms and look for squares and cubes: two terms, identity; three, perfect square or split; four, group.

Can grouping be done in different ways?

Yes; try another arrangement if the brackets do not match.

Is a⁶ − b⁶ in the syllabus?

It is an extension that combines difference of squares and cubes; it is a good test of the identities.