Learn · Mathematics · Class 9

Algebraic Manipulation & Factorization

Expanding, factorising and simplifying, the technique the whole syllabus depends on.

11 parts

Scroll to begin ↓

📘 DefinitionWhy factorising matters mostPart 1 of 11

🧩 The partsThe four methods, in orderPart 2 of 11

🔍 In detailThe standard identitiesPart 3 of 11

✏️ Worked exampleFactorising a trinomial with a leading coefficientPart 4 of 11

✏️ Worked exampleFactorising by groupingPart 5 of 11

🔍 In detailAlgebraic fractionsPart 6 of 11

✏️ Worked exampleChanging the subject of a formulaPart 7 of 11

🌍 In the real worldWhere this is neededPart 8 of 11

⚠️ Common mistakesWhere marks are lostPart 9 of 11

✅ SummaryWhat to rememberPart 10 of 11

✏️ Worked exampleWatch one worked throughPart 11 of 11

1📘 DefinitionPart 1 / 11

Why factorising matters most

Factorising writes an expression as a product of factors, the reverse of expanding.

Why it is the central technique: a product equals zero only if one of its factors is zero.

(x − 3)(x + 5) = 0
→ x = 3 or x = −5

That is how quadratic equations are solved, how algebraic fractions simplify, and how curves are sketched.

Weakness here shows up in every later topic.

2🧩 The partsPart 2 / 11

The four methods, in order

Always check them in this order:

Common factor, take out the HCF. 6x² + 9x = 3x(2x + 3)
Difference of two squares: two terms, a minus, both squares. x² − 16 = (x+4)(x−4)
Trinomial: three terms, x² + bx + c.
Grouping: four terms, taken in pairs.

Step 1 is skipped constantly. 2x² − 18 does not look like a difference of squares until the 2 comes out: 2(x² − 9) = 2(x+3)(x−3).

3🔍 In detailPart 3 / 11

The standard identities

(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

The middle term in the cubic factors takes the opposite sign from the bracket, a plus outside gives a minus inside. Mixing them up is a common error.

Note: a² + b² does not factorise over the reals, though a³ + b³ does. That asymmetry surprises people.

4✏️ Worked examplePart 4 / 11

Factorising a trinomial with a leading coefficient

Question. Factorise 6x² + 11x − 10.

The AC method, step by step.

Step 1: multiply a × c.

6 × (−10) = −60

Step 2: find two numbers multiplying to −60 and adding to +11.

15 and −4 (15 × −4 = −60, 15 + (−4) = 11) ✅

Step 3: split the middle term using them.

6x² + 15x − 4x − 10

Step 4: group in pairs and factorise each.

3x(2x + 5) − 2(2x + 5)

Step 5: take out the common bracket.

= (2x + 5)(3x − 2)

Check by expanding: 6x² − 4x + 15x − 10 = 6x² + 11x − 10 ✅

If the brackets in step 4 do not match, your two numbers were wrong.

5✏️ Worked examplePart 5 / 11

Factorising by grouping

Question. Factorise ax + ay + bx + by.

Step 1: pair the terms sensibly. Choose pairs sharing a factor.

(ax + ay) + (bx + by)

Step 2: factorise each pair.

a(x + y) + b(x + y)

Step 3: the bracket is now a common factor.

= (x + y)(a + b)

The diagnostic: if the two brackets in step 2 are not identical, you paired the terms wrongly. Try a different pairing.

Watch signs carefully when a pair begins with a minus:

2x − 6 − ax + 3a
= 2(x − 3) − a(x − 3)
= (x − 3)(2 − a)

Factoring out −a flips both signs inside its bracket, miss that and the brackets will not match.

6🔍 In detailPart 6 / 11

Algebraic fractions

Simplifying requires factorising first, then cancelling common factors.

(x² − 9)/(x² + 7x + 12)
= (x+3)(x−3) / (x+3)(x+4)
= (x − 3)/(x + 4)

The rule that prevents the worst error: you may cancel factors, never terms.

(x + 3)/(x + 4) , cannot cancel the x
because they are terms, not factors

Adding fractions needs a common denominator, exactly as with numbers:

1/x + 1/y = (y + x)/xy

Dividing: flip the second and multiply.

7✏️ Worked examplePart 7 / 11

Changing the subject of a formula

Question. Make r the subject of A = πr²h.

Undo the operations in reverse order. r was squared, then multiplied by π and h.

Step 1: divide both sides by πh.

A/(πh) = r²

Step 2: take the square root.

r = √(A/πh)

A harder case: the subject appears twice. Make x the subject of ax = bx + c.

Gather the x terms on one side, then factorise:

ax − bx = c
x(a − b) = c
x = c/(a − b)

Factorising is what makes it possible. Whenever the subject appears more than once, collect and factorise.

8🌍 In the real worldPart 8 / 11

Where this is needed

Quadratic equations: factorising is the fastest solution method when it works.

Rearranging formulas in physics and chemistry constantly requires exactly these steps.

Simplifying before substituting reduces arithmetic and therefore error.

Graph sketching. The factorised form gives the x-intercepts immediately.

In the exam: "Simplify fully" and "factorise completely" both mean the work is not finished until nothing further can be extracted. Partial factorisation loses marks.

9⚠️ Common mistakesPart 9 / 11

Where marks are lost

Not taking out the common factor first.
Cancelling terms instead of factors.
Sign errors when factorising out a negative.
Trying to factorise a² + b².
Stopping at partial factorisation when "completely" was asked.
Forgetting the ± when taking a square root in an equation.

10✅ SummaryPart 10 / 11

What to remember

Check in order: common factor → difference of squares → trinomial → grouping.
AC method: split the middle term, then group.
Cancel factors, never terms.
Subject appearing twice → collect and factorise.

11✏️ Worked examplePart 11 / 11

Watch one worked through

Here is one done slowly, with the reason for every move. Then one for you.

Watch one first

Factorise x² - 5x + 6, then factorise 9y² - 16.

For the quadratic, look for two numbers that multiply to +6 and add to -5.
product = +6 sum = −5
Product from the last term, sum from the middle. Writing both targets down before guessing makes the search short.
Both negative, since the product is positive and the sum negative.
−2 × −3 = +6 −2 + −3 = −5 ✓
Reading the signs first halves the work. A positive product with a negative sum can only be two negatives.
Write the brackets and check by expanding.
(x − 2)(x − 3) check: x² −3x −2x +6 = x² −5x +6 ✓
Expanding back takes seconds and is the only way to be certain. Never leave a factorisation unchecked.
The second is a difference of two squares.
9y² − 16 = (3y)² − 4² = (3y + 4)(3y − 4)
Recognise the shape: two perfect squares with a minus between them and no middle term. a² − b² = (a+b)(a−b) every time.

Answer: (x - 2)(x - 3) and (3y + 4)(3y - 4)

Now you try

Factorise x² - 9. Write it as two brackets.

Show the answer

(x+3)(x-3)

Practise Algebraic Manipulation & Factorization

Everything on QuizByt for Class 9 Mathematics that goes with this lesson: read it first, then test yourself while it is fresh.

🧠 5 quizzes on Algebraic Manipulation & Factorization (2 easy, 2 medium, 1 hard)

Work up from easy to hard: the hard quiz asks the Class 9 exam-style questions on Algebraic Manipulation & Factorization.

🖨️ Printable worksheet

Algebraic Manipulation & Factorization on paper for Class 9, with room to write the working out. Preview the first page free.

Algebraic Manipulation & Factorization: Class 9 Mathematics Worksheet

🎮 Games for Class 9 Mathematics

Number and logic games at the right level for Class 9. Passing a quiz on Algebraic Manipulation & Factorization earns the credits that open them.

Now test yourself

You have read the whole lesson. Take the quiz on this topic to see how much stuck,a free account is all it takes.

📊 Your score is saved, so you can see progress over time
🔍 See exactly which questions you got wrong, and why
💎 Earn credits for every quiz you pass, and spend them on games

Create a free account →I already have one

In a hurry? Try a free Mathematics quiz with no account.