DEFINITION

Factorization is the process of breaking an algebraic expression into smaller parts called factors. When you multiply these factors together, you get will the original expression again.

Think of it like breaking a chocolate bar into small pieces. Each piece is a factor, and when you put them together, you will get the full chocolate bar again.

1. It helps us to simplify expressions.


2. It makes solving equations easier.


3. It helps us to understand patterns in numbers and letters.

1. FACTORING OUT THE GREATEST COMMON FACTOR (GCF)

The GCF is the biggest number and/or letter that is common in all terms.

Example 1: Factorize 12x + 8

Step by step:

1. Find GCF of numbers 12 and 8 → GCF = 4


2. Check letters: x is in the first term only, so only number 4 is common.


3. Write each term as 4 × something: 12x = 4 × 3x, 8 = 4 × 2


4. Factor out 4: 12x + 8 = 4(3x + 2)


5. Answer: 4(3x + 2)


6. Explanation: 4 is the common factor, and inside the bracket is what is left after dividing each term by 4.

2. FACTORING OUT A COMMON VARIABLE

Sometimes letters are common too.

Example 2: Factorize 6ab + 9a

1. GCF of numbers 6 and 9 = 3


2. Common letters: a is in both terms. So GCF = 3a


3. Write each term: 6ab = 3a × 2b, 9a = 3a × 3


4. Factor out 3a: 6ab + 9a = 3a(2b + 3)


5. Answer: 3a(2b + 3)

3. FACTORING BY GROUPING

Group terms in pairs or sets that have something in common.

Example 3: Factorize ax + ay + bx + by

1. Group terms: (ax + ay) + (bx + by)


2. Factor each group: ax + ay = a(x + y), bx + by = b(x + y)


3. Factor out the common bracket (x + y): ax + ay + bx + by = (a + b)(x + y)


4. Answer: (a + b)(x + y)

4. FACTORING SIMPLE QUADRATICS

Quadratic expressions are like x² + 5x + 6. We look for two numbers that multiply to the last number (constant) and add to the middle number (coefficient of x).

Example 4: Factorize x² + 5x + 6

1. Find two numbers that multiply to 6 and add to 5 → numbers = 2 and 3


2. Split middle term: x² + 2x + 3x + 6


3. Group terms: (x² + 2x) + (3x + 6)


4. Factor each group: x(x + 2) + 3(x + 2)


5. Factor out common bracket (x + 2) → (x + 2)(x + 3)


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

5. DIFFERENCE OF SQUARES

Formula: A² − B² = (A − B)(A + B)

Example 5: Factorize 9y² − 16

1. Recognize squares: 9y² = (3y)², 16 = 4²


2. Apply formula: (3y − 4)(3y + 4)


3. Answer: (3y − 4)(3y + 4)

6. FACTORING TRINOMIALS WITH LEADING COEFFICIENT NOT 1

Example 6: Factorize 6x² + 11x + 3

1. Multiply first and last numbers: 6 × 3 = 18


2. Find two numbers that multiply to 18 and add to 11 → 9 and 2


3. Split middle term: 6x² + 9x + 2x + 3


4. Group terms: (6x² + 9x) + (2x + 3)


5. Factor each group: 3x(2x + 3) + 1(2x + 3)


6. Factor out common bracket (2x + 3) → (2x + 3)(3x + 1)


7. Answer: (2x + 3)(3x + 1)

7. FACTORING PERFECT SQUARE TRINOMIALS

Formula: A² + 2AB + B² = (A + B)²

Example: Factorize x² + 6x + 9

1. Recognize pattern: x² = x², 9 = 3², 6x = 2×x×3


2. Apply formula: (x + 3)²


3. Answer: (x + 3)²

SHORT REVIEW

• Addition & subtraction → combine like terms first


• Multiplication → multiply numbers and add powers of letters


• Division → divide numbers and subtract powers of letters


• Factorization → break into smaller pieces (factors) that can multiply back to the original expression

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1. FACTORIZATION OF EXPRESSIONS OF THE FORM ax + ay

Definition

Factorization means taking out a common factor from all the terms in an expression so that you can write it as multiplication.

Rule

ax + ay = a(x + y)

Explanation

If two terms share the same letter (or number), you can “take it out” and write the remaining parts in a bracket.

Exam-Style Questions

Example 1

Factorize: 6x + 9x

Solution (step by step):

1. Identify common factor of numbers: 6 and 9 → 3.


2. Factor out 3: 6x = 3(2x), 9x = 3(3x).


3. Combine: 3(2x + 3x) = 3(5x) = 15x.

Final answer: 15x

Example 2

Factorize: 8a + 12b

1. Common number: 4.


2. 8a = 4(2a), 12b = 4(3b).


3. Combine: 4(2a + 3b).

Final answer: 4(2a + 3b)

Example 3

Factorize: 7x + 14y

1. Common number: 7.


2. 7x = 7(x), 14y = 7(2y).


3. Combine: 7(x + 2y).

Final answer: 7(x + 2y)

Example 4

Factorize: 15m + 10n

1. Common number: 5.


2. 15m = 5(3m), 10n = 5(2n).


3. Combine: 5(3m + 2n).

Final answer: 5(3m + 2n)

Example 5

Factorize: 12p + 18q

1. Common number: 6.


2. 12p = 6(2p), 18q = 6(3q).


3. Combine: 6(2p + 3q).

Final answer: 6(2p + 3q)

Example 6

Factorize: 20x + 30y

1. Common number: 10.


2. 20x = 10(2x), 30y = 10(3y).


3. Combine: 10(2x + 3y).

Final answer: 10(2x + 3y)

Example 7

Factorize: 14a + 21b

1. Common number: 7.


2. 14a = 7(2a), 21b = 7(3b).


3. Combine: 7(2a + 3b).

Final answer: 7(2a + 3b)

Example 8

Factorize: 9x + 12y

1. Common number: 3.


2. 9x = 3(3x), 12y = 3(4y).


3. Combine: 3(3x + 4y).

Final answer: 3(3x + 4y)

Example 9

Factorize: 25m + 35n

1. Common number: 5.


2. 25m = 5(5m), 35n = 5(7n).


3. Combine: 5(5m + 7n).

Final answer: 5(5m + 7n)

Example 10

Factorize: 18p + 24q

1. Common number: 6.


2. 18p = 6(3p), 24q = 6(4q).


3. Combine: 6(3p + 4q).

Final answer: 6(3p + 4q)

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2. FACTORIZATION OF EXPRESSIONS OF THE FORM 3m + pq + 3p + mp

Definition

Here we group terms cleverly to find common factors. This method is called factorization by grouping.

Rule

Group terms in pairs so that each pair has a common factor, then factor out the common factors and combine.

Example (illustration)

Expression: 3m + pq + 3p + mp

Step 1: Group terms: (3m + 3p) + (pq + mp)

Step 2: Factor each group:
3m + 3p = 3(m + p)
pq + mp = p(q + m) = p(m + q)

Note: The bracket orders must match to factor further. If they do not match you may try another grouping such as (3m + mp) + (3p + pq):

3m + mp = m(3 + p)
3p + pq = p(3 + q)

Sometimes rearrangement helps; other times expression stops there.

Exam-Style Questions

1. Factorize: 3m + 3p + m + p → (3m + m) + (3p + p) = 4m + 4p = 4(m + p)


2. Factorize: 5a + 5b + 2a + 2b → (5a + 2a) + (5b + 2b) = 7a + 7b = 7(a + b)


3. Factorize: 6x + 9y + 2x + 3y → (6x + 2x) + (9y + 3y) = 8x + 12y = 4(2x + 3y)


4. Factorize: 2m + 2n + 3m + 3n → (2m + 3m) + (2n + 3n) = 5m + 5n = 5(m + n)


5. Factorize: 7p + 3q + 14p + 6q → (7p + 14p) + (3q + 6q) = 21p + 9q = 3(7p + 3q)


6. Factorize: 4x + 2y + 8x + 6y → (4x + 8x) + (2y + 6y) = 12x + 8y = 4(3x + 2y)


7. Factorize: 5a + 10b + 15a + 20b → (5a + 15a) + (10b + 20b) = 20a + 30b = 10(2a + 3b)


8. Factorize: 3m + 9n + 6m + 12n → (3m + 6m) + (9n + 12n) = 9m + 21n = 3(3m + 7n)


9. Factorize: 8x + 4y + 12x + 6y → (8x + 12x) + (4y + 6y) = 20x + 10y = 10(2x + y)


10. Factorize: 6p + 9q + 12p + 18q → (6p + 12p) + (9q + 18q) = 18p + 27q = 9(2p + 3q)

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3. FACTORIZATION OF EXPRESSIONS OF THE FORM a − b² (Difference of Squares)

Definition

This is called difference of squares. It happens when you have one square number minus another square number.

Rule

A2 − B2 = (A − B)(A + B)

Example

Factorize: x² − 9

1. Recognize squares: x² = x × x, 9 = 3 × 3.


2. Apply formula: x² − 3² = (x − 3)(x + 3).

Final answer: (x − 3)(x + 3)

Examples

1. x² − 16 = (x − 4)(x + 4)


2. 9y² − 25 = (3y − 5)(3y + 5)


3. 4a² − 49 = (2a − 7)(2a + 7)


4. 36m² − 64 = (6m − 8)(6m + 8)


5. x² − 1 = (x − 1)(x + 1)


6. 25p² − 9q² = (5p − 3q)(5p + 3q)


7. 49x² − 64y² = (7x − 8y)(7x + 8y)


8. 100a² − 121b² = (10a − 11b)(10a + 11b)


9. 81x² − 36 = (9x − 6)(9x + 6)


10. 4m² − 49n² = (2m − 7n)(2m + 7n)


11. x² − 36 = (x − 6)(x + 6)


12. 16y² − 1 = (4y − 1)(4y + 1)


13. 25x² − 64 = (5x − 8)(5x + 8)


14. 49a² − 1 = (7a − 1)(7a + 1)


15. 9p² − 4q² = (3p − 2q)(3p + 2q)


16. 36x² − 81 = (6x − 9)(6x + 9)


17. 64y² − 49 = (8y − 7)(8y + 7)


18. 121a² − 4b² = (11a − 2b)(11a + 2b)


19. 100x² − 25y² = (10x − 5y)(10x + 5y)


20. 9m² − 1 = (3m − 1)(3m + 1)

• Always look for squares.


• Apply A2 − B2 = (A − B)(A + B).


• Keep it short and clean; the exam likes direct answers.

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4. FACTORIZATION OF EXPRESSIONS OF THE FORM a² − ab + b

Definition

This is a trinomial with three terms. We first look for common factors, then try grouping or the split-middle-term method where applicable.

Explanation

Sometimes a2 − ab + b can be partially factored (take out common parts). In many JSSCE examples the trinomial is chosen so splitting the middle term works (e.g. when the constant is simple).

Short Example (illustration)

Expression: x² − xy + y

1. Group the first two terms: (x² − xy) + y


2. Factor the first group: x(x − y)


3. The second term y has no (x − y) factor, so expression stays as: x(x − y) + y

This is a partial factor form: x(x − y) + y. Unless the expression has a nicer pattern, we stop here.

Practice Questions Examples

1. x² − 5x + 6 = (x − 2)(x − 3)


2. x² − 7x + 12 = (x − 3)(x − 4)


3. x² − 8x + 15 = (x − 3)(x − 5)


4. x² − 6x + 8 = (x − 2)(x − 4)


5. x² − 9x + 20 = (x − 4)(x − 5)


6. x² − 11x + 24 = (x − 3)(x − 8)


7. x² − 10x + 21 = (x − 3)(x − 7)


8. x² − 12x + 35 = (x − 5)(x − 7)


9. x² − 13x + 40 = (x − 5)(x − 8)


10. x² − 14x + 45 = (x − 5)(x − 9)


11. x² − 15x + 56 = (x − 7)(x − 8)


12. x² − 16x + 60 = (x − 10)(x − 6)


13. x² − 17x + 72 = (x − 9)(x − 8)


14. x² − 18x + 77 = (x − 11)(x − 7)


15. x² − 19x + 90 = (x − 10)(x − 9)


16. x² − 20x + 96 = (x − 12)(x − 8)


17. x² − 21x + 110 = (x − 11)(x − 10)


18. x² − 22x + 120 = (x − 10)(x − 12)


19. x² − 23x + 132 = (x − 11)(x − 12)


20. x² − 24x + 135 = (x − 9)(x − 15)

Tips:

• Look for two numbers that multiply to the last number (constant) and add/subtract to the middle coefficient.


• Use splitting the middle term method and factor by grouping.


• Write the final answer as two brackets.

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WORD PROBLEMS INVOLVING FACTORIZATION

Definition:

Factorization means finding what is common in all the terms of an expression and writing it outside brackets. Think of it like sharing candies equally among friends: we see what each friend can get and put the rest in a group.

1. Amina has x² + 5x beads. Factorize.



Step 1: Look at each term: x² and 5x.

Step 2: Ask: What is common in x² and 5x? → Both have x.

Step 3: Take x outside the bracket.

Step 4: Inside bracket, divide each term by x: x² ÷ x = x, 5x ÷ x = 5.

Step 5: Write the final answer: x(x + 5)

2. A farmer planted 6ab + 9a seeds. Factorize.



Step 1: Look at each term: 6ab and 9a.

Step 2: Find common number: 6 and 9 → common factor = 3

Step 3: Find common letters: both have a → take a too.

Step 4: Take 3a outside bracket.

Step 5: Inside bracket: 6ab ÷ 3a = 2b, 9a ÷ 3a = 3.

Step 6: Final answer: 3a(2b + 3)

3. A shop sold 3x + 3y items. Factorize.



Step 1: Look at 3x and 3y.

Step 2: Common number: 3. Both terms have no letters in common.

Step 3: Take 3 outside.

Step 4: Inside: 3x ÷ 3 = x, 3y ÷ 3 = y.

Step 5: Answer: 3(x + y)

4. A rectangle has length 2x + 4. Factorize.



Step 1: Look at 2x and 4.

Step 2: Common number = 2

Step 3: Take 2 outside.

Step 4: Inside: 2x ÷ 2 = x, 4 ÷ 2 = 2.

Step 5: Answer: 2(x + 2)

5. The sum of two numbers is written as x² + xy. Factorize.



Step 1: Look at x² and xy.

Step 2: Common letter = x.

Step 3: Take x outside.

Step 4: Inside: x² ÷ x = x, xy ÷ x = y.

Step 5: Answer: x(x + y)

6. The total cost of pencils is written as 5m + 10n. Factorize.



Step 1: Look at 5m and 10n.

Step 2: Common number = 5.

Step 3: Take 5 outside.

Step 4: Inside: 5m ÷ 5 = m, 10n ÷ 5 = 2n.

Step 5: Answer: 5(m + 2n)

7. A gardener has 3x + 3y + 6 plants. Factorize.



Step 1: Look at 3x, 3y, and 6.

Step 2: Common number = 3.

Step 3: Take 3 outside.

Step 4: Inside: 3x ÷ 3 = x, 3y ÷ 3 = y, 6 ÷ 3 = 2.

Step 5: Answer: 3(x + y + 2)

8. A factory produced 4ab + 6ac items. Factorize.



Step 1: Look at 4ab and 6ac.

Step 2: Common number = 2.

Step 3: Common letter = a.

Step 4: Take 2a outside.

Step 5: Inside: 4ab ÷ 2a = 2b, 6ac ÷ 2a = 3c.

Step 6: Answer: 2a(2b + 3c)

9. The sum of two expressions is written as x² + 3x. Factorize.



Step 1: Look at x² and 3x.

Step 2: Common letter = x.

Step 3: Take x outside.

Step 4: Inside: x² ÷ x = x, 3x ÷ x = 3.

Step 5: Answer: x(x + 3)

10. A car rental company charges 6x + 9. Factorize.



Step 1: Look at 6x and 9.

Step 2: Common number = 3.

Step 3: Take 3 outside.

Step 4: Inside: 6x ÷ 3 = 2x, 9 ÷ 3 = 3.

Step 5: Answer: 3(2x + 3)