IB Class 9 Algebra Expansion and Factorisation Questions

IB Class 9 Algebra Expansion and Factorisation Questions

IB Class 9 Algebra Expansion and Factorisation Questions

IB Class 9 Algebra Expansion and Factorisation Questions help students master polynomial expansion and identity-based simplification. These exercises focus on applying algebraic formulas accurately. Moreover, learners develop logical reasoning skills through structured practice. Clear step-by-step examples improve understanding.

Build Strong Algebra Foundations

IB Class 9 Algebra Expansion and Factorisation Questions support effective revision before exams. Therefore, students gain confidence while solving complex expressions. Additionally, consistent practice increases calculation speed. As a result, learners perform better in assessments. Strong fundamentals ensure long-term academic success.

Expansion and Factorisation Practice Resources

Students can download IB Math Class 9 Expansion and Factorisation Worksheet PDF for structured revision. Moreover, IB Math Expansion and Factorisation Practice Problems PDF provide additional exercises. Therefore, regular practice strengthens algebra skills and improves exam performance efficiently.

Expansion and Factorisation

Expansion and factorisation are inverse processes in algebra. Expanding means removing brackets by multiplying, while factorisation involves writing an expression as a product of its factors. These skills are essential for solving equations, simplifying expressions, and understanding algebraic relationships.

1. Expansion (Removing Brackets)

Expansion is the process of multiplying out brackets. It is based on the distributive law: a(b + c) = ab + ac.

1.1 Single Brackets

Multiply the term outside the bracket by each term inside.

Example 1: Expand 3(2x + 5).

3(2x + 5) = 3 × 2x + 3 × 5 = 6x + 15

Example 2: Expand -4(3y – 7).

-4(3y – 7) = -4 × 3y + (-4) × (-7) = -12y + 28

1.2 Double Brackets (FOIL Method)

To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second bracket. FOIL stands for First, Outer, Inner, Last.

(a + b)(c + d) = ac + ad + bc + bd

Example 1: Expand (x + 4)(x + 7).

(x + 4)(x + 7) = x·x + x·7 + 4·x + 4·7
= x² + 7x + 4x + 28 = x² + 11x + 28

Example 2: Expand (2y – 3)(y + 5).

(2y – 3)(y + 5) = 2y·y + 2y·5 + (-3)·y + (-3)·5
= 2y² + 10y – 3y – 15 = 2y² + 7y – 15

1.3 Perfect Squares

These are special cases of double brackets worth memorising.

(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²

Example 1: Expand (x + 5)².

(x + 5)² = x² + 2·x·5 + 5² = x² + 10x + 25

Example 2: Expand (3y – 2)².

(3y – 2)² = (3y)² – 2·(3y)·2 + 2² = 9y² – 12y + 4

1.4 Difference of Two Squares

This is another special product pattern.

(a + b)(a – b) = a² – b²

Example 1: Expand (x + 3)(x – 3).

(x + 3)(x – 3) = x² – 9

Example 2: Expand (2y + 7)(2y – 7).

(2y + 7)(2y – 7) = (2y)² – 7² = 4y² – 49

1.5 More Than Two Brackets

Expand step by step, multiplying two brackets at a time.

Example: Expand (x + 1)(x + 2)(x + 3).

First, (x + 1)(x + 2) = x² + 3x + 2
Then multiply by (x + 3): (x² + 3x + 2)(x + 3)
= x³ + 3x² + 2x + 3x² + 9x + 6
= x³ + 6x² + 11x + 6

2. Factorisation (Putting Brackets Back)

Factorisation is the reverse of expansion. We write an expression as a product of its factors. It’s like finding what was multiplied together to get the expression.

2.1 Common Factors (Highest Common Factor)

Identify the greatest factor common to all terms and factor it out.

Example 1: Factorise 6x + 9.

HCF of 6 and 9 is 3. 6x + 9 = 3(2x + 3)

Example 2: Factorise 12y² – 8y.

HCF of 12 and 8 is 4, and both terms have y. So HCF = 4y.
12y² – 8y = 4y(3y – 2)

Example 3: Factorise 5x²y – 10xy².

HCF = 5xy. 5x²y – 10xy² = 5xy(x – 2y)

2.2 Factorising by Grouping

Used for expressions with four terms. Group terms in pairs, factor each pair, then factor out the common bracket.

Example 1: Factorise x³ + 3x² + 2x + 6.

Group: (x³ + 3x²) + (2x + 6)
= x²(x + 3) + 2(x + 3)
= (x + 3)(x² + 2)

Example 2: Factorise 2xy – 6x + 5y – 15.

Group: (2xy – 6x) + (5y – 15)
= 2x(y – 3) + 5(y – 3)
= (y – 3)(2x + 5)

3. Factorising Quadratics: x² + bx + c

We want to find two numbers that multiply to give ‘c’ and add to give ‘b’.

Step-by-Step Process

For x² + bx + c, find two numbers p and q such that:

p × q = c   and   p + q = b

Then the factorised form is (x + p)(x + q).

Example 1: Factorise x² + 7x + 12.

Find factors of 12 that add to 7: 3 and 4 (3×4=12, 3+4=7)
= (x + 3)(x + 4)

Example 2: Factorise x² – 5x + 6.

Find factors of 6 that add to -5: (-2) and (-3) [(-2)×(-3)=6, (-2)+(-3)=-5]
= (x – 2)(x – 3)

Example 3: Factorise x² – x – 12.

Find factors of -12 that add to -1: 3 and -4 [3×(-4)=-12, 3+(-4)=-1]
= (x + 3)(x – 4)

4. Factorising Quadratics: ax² + bx + c (a ≠ 1)

This requires a systematic approach. We’ll use the ‘ac method’.

AC Method (Grouping Method)

  1. Multiply a and c to get ac.
  2. Find two numbers that multiply to ac and add to b.
  3. Split the middle term (bx) using these two numbers.
  4. Factor by grouping.

Example 1: Factorise 2x² + 7x + 3.

a=2, c=3 → ac = 6. Find factors of 6 that add to 7: 1 and 6.
Rewrite: 2x² + x + 6x + 3
Group: (2x² + x) + (6x + 3) = x(2x + 1) + 3(2x + 1)
= (2x + 1)(x + 3)

Example 2: Factorise 3x² – 10x – 8.

ac = 3 × (-8) = -24. Find factors of -24 that add to -10: -12 and 2.
Rewrite: 3x² – 12x + 2x – 8
Group: (3x² – 12x) + (2x – 8) = 3x(x – 4) + 2(x – 4)
= (x – 4)(3x + 2)

Alternative: Trial and Error

Sometimes you can guess factors by considering the factors of a and c.

Example: Factorise 2x² – 5x – 12.

Possible factors of 2x²: 2x and x. Factors of -12: (-3,4), (3,-4), (-2,6), (2,-6), (-1,12), (1,-12)
Try (2x + 3)(x – 4) = 2x² – 8x + 3x -12 = 2x² -5x -12 ✓
= (2x + 3)(x – 4)

5. Special Factorisation Forms

5.1 Difference of Two Squares

a² – b² = (a + b)(a – b)

Example 1: Factorise x² – 25.

= (x + 5)(x – 5)

Example 2: Factorise 4y² – 81.

= (2y)² – 9² = (2y + 9)(2y – 9)

Example 3: Factorise (x+2)² – 16.

Treat (x+2) as ‘a’ and 4 as ‘b’.
= [(x+2) + 4][(x+2) – 4] = (x + 6)(x – 2)

5.2 Perfect Square Trinomials

a² + 2ab + b² = (a + b)²
a² – 2ab + b² = (a – b)²

Example 1: Factorise x² + 8x + 16.

= x² + 2·x·4 + 4² = (x + 4)²

Example 2: Factorise 9y² – 30y + 25.

= (3y)² – 2·(3y)·5 + 5² = (3y – 5)²

5.3 Sum and Difference of Cubes (Extension)

For advanced students, these patterns are useful.

a³ + b³ = (a + b)(a² – ab + b²)
a³ – b³ = (a – b)(a² + ab + b²)

Example: Factorise x³ + 27.

= x³ + 3³ = (x + 3)(x² – 3x + 9)

6. Using Expansion and Factorisation Together

Often, problems require both skills, such as simplifying algebraic fractions or solving quadratic equations.

Solved Examples

Example 1 (Simplify fraction): Simplify (x² – 9) / (x + 3).

Factorise numerator: x² – 9 = (x + 3)(x – 3)
So (x + 3)(x – 3) / (x + 3) = x – 3 (provided x ≠ -3)

Example 2 (Solve equation): Solve x² – 5x + 6 = 0.

Factorise: (x – 2)(x – 3) = 0
Therefore x = 2 or x = 3

Example 3 (Expand then factorise): Expand (x + 2)² – (x – 3)² and simplify.

Expand: (x² + 4x + 4) – (x² – 6x + 9)
= x² + 4x + 4 – x² + 6x – 9 = 10x – 5
Factorise: = 5(2x – 1)

Example 4 (Area problem): A square of side x has a smaller square of side 3 cut from its corner. Write the remaining area in factorised form.

Area = x² – 9 = (x + 3)(x – 3)

7. Common Pitfalls

  • Incorrect: (a + b)² = a² + b².
    ✓ Correct: (a + b)² = a² + 2ab + b². Always include the middle term.
  • Incorrect: Forgetting the sign when expanding with negatives: (x – 4)(x – 2) = x² – 6x – 8?
    ✓ Correct: (x – 4)(x – 2) = x² – 2x – 4x + 8 = x² – 6x + 8.
  • Incorrect: Thinking a² – b² = (a – b)².
    ✓ Correct: (a – b)² = a² – 2ab + b², which is different from a² – b².
  • Incorrect: Leaving out the HCF when factorising (e.g., 4x + 6 = 2(2x + 3) is correct, but 2(2x+3) is fully factorised? Actually, it is fine, but if you miss a common factor, it’s incomplete.
  • Incorrect: For the ac method, forgetting to split the middle term correctly or not grouping properly.
  • Incorrect: Cancelling terms instead of factors in fractions: (x² + 3x)/(x) = x² + 3? No! Factorise first: x(x+3)/x = x+3 (if x≠0).

8. Practice Questions

  1. Expand and simplify: (2x + 3)(x – 5).
  2. Expand: (3y – 4)².
  3. Factorise completely: 15x²y – 10xy².
  4. Factorise: x² – 8x + 15.
  5. Factorise: 6x² + 11x – 10.
  6. Factorise: 49a² – 100b².
  7. Simplify the algebraic fraction: (x² – 4x – 12)/(x – 6).
  8. Solve by factorising: 2x² + 5x – 3 = 0.

Answers: 1) 2x² -7x -15   2) 9y² -24y +16   3) 5xy(3x – 2y)   4) (x – 3)(x – 5)   5) (2x + 5)(3x – 2)   6) (7a + 10b)(7a – 10b)   7) x + 2 (x≠6)   8) x = 1/2 or x = -3

Expansion and Factorisation: Word Problems (Level 1)

Question 1: A rectangular garden has length (3x + 4) m and width (2x – 1) m. Find an expanded expression for the area. If x = 5, calculate the actual area. Answer: Area = 6x2 + 5x – 4 m2; If x = 5, Area = 171 m2
Question 2: A number when multiplied by (x + 3) gives (x2 + 7x + 12). Find the number. Answer: (x + 4)
Question 3: The product of two consecutive integers is 56 more than the square of the smaller integer (n). Form an equation and find the integers. Answer: n(n+1) = n2 + 56; Integers are 56 and 57
Question 4: A square cardboard of side (2x + 3) cm has a small square of side x cm cut from each corner. Write the remaining area in factorised form. Answer: 3(4x + 3) square cm (Note: Logic corrected from source)
Question 5: The sum of two numbers is 15 and their product is 56. Let one be ‘a’. Form a quadratic equation and find the numbers. Answer: a2 – 15a + 56 = 0; Numbers are 7 and 8
Question 6: A pool (x + 6)m by (x – 2)m is surrounded by a path of uniform width x m. Find the total area expression. Answer: 9x2 + 12x – 12 m2
Question 7: Difference between two numbers is 4 and sum of squares is 136. If smaller is y, find the numbers. Answer: (y+4)2 + y2 = 136; Numbers are 6 and 10
Question 8: Area of a field is (x2 + 5x – 24). If length is (x + 8), find the width. Answer: Width = (x – 3)
Question 9: (x + a)(x + b) = x2 + 11x + 30. Find positive integers a and b (where a > b). Answer: a = 6, b = 5
Question 10: A number (p) plus its reciprocal is 41/20. Form a quadratic and find p. Answer: 20p2 – 41p + 20 = 0; p = 5/4 or 4/5
Question 11: Length is 3 cm more than twice its width. Area is 65 cm2. Find dimensions. Answer: Width = 5 cm, Length = 13 cm
Question 12: Factorise: x2 + (5/6)x + (1/6) Answer: (x + 1/2)(x + 1/3)
Question 13: Sum of squares of two consecutive odd numbers is 202. Find the numbers. Answer: 9 and 11
Question 14: Volume of a box is (x3 + 3x2 – 4x – 12). Height is (x – 2) and width (x + 2). Find length. Answer: (x + 3) cm
Question 15: A garden (12m x 16m) has a path of width x m around it. Total area is 285 m2. Find x. Answer: 4x2 + 56x – 93 = 0; x = 1.5 m

Step-by-Step Solutions

Sol 1: (3x+4)(2x-1) = 6x2 – 3x + 8x – 4 = 6x2+5x-4. Sub x=5: 6(25)+5(5)-4 = 171.
Sol 2: Factorise x2+7x+12 into (x+3)(x+4). Since one factor is (x+3), the other is (x+4).
Sol 3: n(n+1) = n2+56 → n2+n = n2+56 → n=56. Integers: 56, 57.
Sol 4: Area = (2x+3)2 – 4x2 = (4x2+12x+9) – 4x2 = 12x + 9 = 3(4x+3).
Sol 5: a(15-a)=56 → 15a-a2=56 → a2-15a+56=0. (a-7)(a-8)=0. Numbers: 7, 8.
Sol 6: New length: (x+6) + 2x = 3x+6. New width: (x-2) + 2x = 3x-2. Area = (3x+6)(3x-2) = 9x2+12x-12.
Sol 7: y2+(y+4)2=136 → 2y2+8y-120=0 → y2+4y-60=0. (y+10)(y-6)=0. y=6.
Sol 8: (x2+5x-24)/(x+8) = (x+8)(x-3)/(x+8) = x-3.
Sol 9: Sum = 11, Product = 30. Factors are 6 and 5. Since a>b, a=6, b=5.
Sol 10: p + 1/p = 41/20 → 20p2-41p+20=0. (5p-4)(4p-5)=0. p = 4/5 or 5/4.
Sol 11: w(2w+3)=65 → 2w2+3w-65=0 → (2w+13)(w-5)=0. w=5, l=13.
Sol 12: 1/6(6x2+5x+1) = 1/6(3x+1)(2x+1) = (x+1/3)(x+1/2).
Sol 13: n2+(n+2)2=202 → 2n2+4n-198=0 → n2+2n-99=0. (n+11)(n-9)=0. n=9.
Sol 14: Vol = (x+3)(x2-4). Divide by (x+2)(x-2) which is (x2-4). Result: x+3.
Sol 15: (16+2x)(12+2x)=285 → 4x2+56x-93=0. x = 1.5 using quadratic formula.

Expansion and Factorisation: MCQ Test (Level 2)

Question 1: The area of a rectangular research plot is calculated as (3x + 7)(2x – 5). Which expression represents the expanded form?
  • (A) 6x2 – x – 35
  • (B) 6x2 – 15x + 14x – 35
  • (C) 6x2 – x + 35
  • (D) 5x2 – x – 35
Correct Option: A
Question 2: A square metal sheet has side length (5x – 4) cm. What is its area in expanded form?
  • (A) 25x2 – 16
  • (B) 25x2 – 40x + 16
  • (C) 10x2 – 40x + 16
  • (D) 25x2 – 20x + 16
Correct Option: B
Question 3: A designer models a pattern using (4x + 3)2. Which is the correct expansion?
  • (A) 16x2 + 24x + 9
  • (B) 16x2 + 12x + 9
  • (C) 8x2 + 24x + 9
  • (D) 16x2 + 9
Correct Option: A
Question 4: The expression for the difference of two squares in a physics model is 9x2 – 25. Which is its factorised form?
  • (A) (3x – 5)2
  • (B) (9x – 5)(x + 5)
  • (C) (3x + 5)(3x – 5)
  • (D) (9x – 25)
Correct Option: C
Question 5: A business profit is represented by (2x – 3)(2x + 3). What is the simplified form?
  • (A) 4x2 – 9
  • (B) 4x2 + 9
  • (C) 2x2 – 9
  • (D) 4x2 – 6x + 9
Correct Option: A
Question 6: A rectangular banner has dimensions (x + 9) and (x – 4). What is the expanded expression for its area?
  • (A) x2 + 5x – 36
  • (B) x2 – 36
  • (C) x2 + 9x – 4x – 36
  • (D) x2 + 13x – 36
Correct Option: A
Question 7: A scientist models growth using (3x – 2)2 – (3x + 2)2. What is the simplified result?
  • (A) -24x
  • (B) 24x
  • (C) -12x
  • (D) 12x
Correct Option: A
Question 8: The volume expression of a model simplifies to x2 + 11x + 28. Which is its factorised form?
  • (A) (x + 7)(x + 4)
  • (B) (x + 14)(x + 2)
  • (C) (x + 28)(x + 1)
  • (D) (x + 8)(x + 3)
Correct Option: A
Question 9: A tile pattern area is represented by (6x + 1)(x – 3). What is the expanded form?
  • (A) 6x2 – 17x – 3
  • (B) 6x2 – 18x + x – 3
  • (C) 6x2 – 3x – 3
  • (D) 6x2 – 17x + 3
Correct Option: A
Question 10: The expression 4x2 – 12x + 9 represents the area of a square. What is its factorised form?
  • (A) (2x – 3)2
  • (B) (4x – 3)2
  • (C) (2x + 3)2
  • (D) (4x + 3)2
Correct Option: A
Question 11: A revenue model simplifies to x2 – 16x + 64. Which is the factorised form?
  • (A) (x – 8)2
  • (B) (x – 4)2
  • (C) (x – 16)2
  • (D) (x + 8)2
Correct Option: A
Question 12: A school garden area is given by (x + 12)2 – 144. What is the simplified expression?
  • (A) x2 + 24x
  • (B) x2 + 144
  • (C) x2 + 24x + 144
  • (D) x2 – 144
Correct Option: A
Question 13: The expression (5x – 6)(x + 2) represents production cost. What is the expanded form?
  • (A) 5x2 + 4x – 12
  • (B) 5x2 + 10x – 6x – 12
  • (C) 5x2 + 4x + 12
  • (D) 5x2 – 4x – 12
Correct Option: A
Question 14: A mathematical model simplifies to 16x2 – 81. Which is the correct factorisation?
  • (A) (4x – 9)2
  • (B) (16x – 9)(x + 9)
  • (C) (4x + 9)(4x – 9)
  • (D) (8x – 9)(2x + 9)
Correct Option: C
Question 15: A designer simplifies (2x + 5)2 + (2x – 5)2. What is the result?
  • (A) 8x2 + 50
  • (B) 4x2 + 50
  • (C) 8x2
  • (D) 4x2 – 50
Correct Option: A

Verifications & Solutions

Sol 1: FOIL: 6x2 – 15x + 14x – 35 = 6x2 – x – 35.
Sol 2: (a-b)2 = a2 – 2ab + b2 → (5x)2 – 2(5x)(4) + 42 = 25x2 – 40x + 16.
Sol 3: (4x)2 + 2(4x)(3) + 32 = 16x2 + 24x + 9.
Sol 4: Difference of squares: (3x)2 – 52 = (3x-5)(3x+5).
Sol 5: (a-b)(a+b) = a2 – b2 → (2x)2 – 32 = 4x2 – 9.
Sol 6: x2 – 4x + 9x – 36 = x2 + 5x – 36.
Sol 7: (9x2 – 12x + 4) – (9x2 + 12x + 4) = -24x.
Sol 8: Factors of 28 that add to 11 are 7 and 4 → (x+7)(x+4).
Sol 9: 6x(x) + 6x(-3) + 1(x) + 1(-3) = 6x2 – 18x + x – 3 = 6x2 – 17x – 3.
Sol 10: Perfect square trinomial: (2x)2 – 2(2x)(3) + 32 = (2x-3)2.
Sol 11: (x)2 – 2(x)(8) + 82 = (x-8)2.
Sol 12: (x2 + 24x + 144) – 144 = x2 + 24x.
Sol 13: 5x2 + 10x – 6x – 12 = 5x2 + 4x – 12.
Sol 14: (4x)2 – 92 = (4x+9)(4x-9).
Sol 15: (4x2 + 20x + 25) + (4x2 – 20x + 25) = 8x2 + 50.

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Frequently Asked Questions (FAQs)

What are IB Class 9 Algebra Expansion and Factorisation Questions?

They provide structured exercises on algebraic identities, expansion, and factorisation with solutions.

How do IB Class 9 Algebra Expansion and Factorisation Questions help students?

They improve accuracy and strengthen algebra fundamentals.

Are worksheets available in PDF format?

Yes, including IB Math Class 9 Expansion and Factorisation Worksheet PDF.

Do these questions follow IB curriculum standards?

Yes, they align with IB learning objectives.

Are algebraic identities covered in these questions?

Yes, identity-based applications are included.

Can students use these questions for exam preparation?

Yes, they support systematic revision.

Are solved examples included?

Yes, step-by-step explanations are provided.

Do these resources include practice problem PDFs?

Yes, IB Math Expansion and Factorisation Practice Problems PDF are available.

How often should students practice expansion and factorisation?

Regular practice improves speed and confidence.

Where can students download structured worksheets?

They can download PDFs for organized revision.