Loading...

About Factorisation — Class 8 Olympiad

Factor algebraic expressions using common factors, identities, and grouping; solve factorization puzzles. This topic is part of the Olympiad Class 8 mathematics syllabus (chapter: Module 7). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Factorisation — solved examples for Class 8 Olympiad

Example 1easy

If x ≠ y, what is the value of (x²y - xy²) / (x - y)?
  1. A)x + y
  2. B)xy
  3. C)x - y
  4. D)x²y²

Step-by-step solution

  1. Factor out the common monomial xy from the numerator:
    x2y−xy2=xy(x−y)x²y - xy² = xy(x - y)
  2. Substitute this back into the expression:
    xy(x−y)/(x−y)xy(x - y) / (x - y)
  3. Since x ≠ y, (x - y) ≠ 0, so we can cancel (x - y) from the numerator and denominator.
    xyxy

Answer: xy

Example 2medium

Factorise the expression (x - y)³ + 3(y - x)² - 10(x - y). Which of the following is a factor?
  1. A)(x - y - 2)
  2. B)(x - y - 3)
  3. C)(x - y + 5)
  4. D)(x - y + 3)

Step-by-step solution

  1. Let A = (x - y). Since (y - x)² = (-(x - y))² = (x - y)², the expression becomes A³ + 3A² - 10A.
  2. Factor out A from the expression: A(A² + 3A - 10).
  3. Factor the quadratic trinomial A² + 3A - 10. We look for two numbers that multiply to -10 and add to 3, which are 5 and -2. So, A² + 3A - 10 = (A + 5)(A - 2).
  4. Substitute back A = (x - y): (x - y)(x - y + 5)(x - y - 2).
  5. From the given options, (x - y - 2) is a factor.

Answer: (x - y - 2)

Example 3hard

Given that a = 2023, b = 2024, and c = 2025, what is the value of (a-b)³ + (b-c)³ + (c-a)³?
  1. A)0
  2. B)3
  3. C)6
  4. D)9

Step-by-step solution

  1. Let x = a-b, y = b-c, and z = c-a.
  2. Calculate the values: x = 2023 - 2024 = -1; y = 2024 - 2025 = -1; z = 2025 - 2023 = 2.
  3. Check the sum: x + y + z = (-1) + (-1) + 2 = 0.
  4. According to the identity, if x + y + z = 0, then x³ + y³ + z³ = 3xyz. So, the expression becomes 3 × (-1) × (-1) × 2 = 6.

Answer: 6

Practice questions on Factorisation

  1. Q1.easy

    Factorize the expression a² + ab + ac + bc.
    1. A)(a+c)(a-b)
    2. B)(a+b)(a+c)
    3. C)(a-b)(a-c)
    4. D)(a+b)(b+c)
    Show answer

    Answer: (a+b)(a+c)

    Hint: Group the terms strategically to find common binomial factors.

  2. Q2.easy

    Factorize x²y + xy² - x - y.
    1. A)(xy-1)(x+y)
    2. B)(xy+1)(x-y)
    3. C)(x-y)(x+y)
    4. D)(x+1)(y-1)
    Show answer

    Answer: (xy-1)(x+y)

    Hint: Try grouping the first two terms and the last two terms separately.

  3. Q3.easy

    For what value of k will 9x² + kx + 25 be a perfect square trinomial?
    1. A)15
    2. B)30
    3. C)±15
    4. D)±30
    Show answer

    Answer: ±30

    Hint: Recall the form of a perfect square trinomial (Ax ± B)² = A²x² ± 2ABx + B².

  4. Q4.medium

    Factorise 4a² - 9b² + 12a + 9. Which of the following is a factor?
    1. A)(2a - 3b + 3)
    2. B)(2a + 3b - 3)
    3. C)(2a + 3b + 9)
    4. D)(2a - 3b - 3)
    Show answer

    Answer: (2a - 3b + 3)

    Hint: Look for a perfect square trinomial hidden within the expression, then apply the difference of squares identity.

  5. Q5.medium

    If x = (a + 1/a) and y = (a - 1/a), where a ≠ 0, find the value of x⁴ - y⁴.
    1. A)4(a² + 1/a²)
    2. B)8(a² + 1/a²)
    3. C)16a²
    4. D)4a²
    Show answer

    Answer: 8(a² + 1/a²)

    Hint: Start by applying the difference of squares identity repeatedly. Then substitute the given values of x and y.

  6. Q6.medium

    If a² + b² = 13 and ab = 6, find the positive value of (a - b) / (a + b).
    1. A)1/5
    2. B)1/3
    3. C)2/5
    4. D)3/5
    Show answer

    Answer: 1/5

    Hint: Recall the identities for (a+b)² and (a-b)² in terms of a² + b² and ab.

  7. Q7.hard

    Evaluate the sum: (100² - 99²) + (98² - 97²) + ... + (2² - 1²).
    1. A)4950
    2. B)5000
    3. C)5025
    4. D)5050
    Show answer

    Answer: 5050

    Hint: Apply the difference of squares identity (a² - b² = (a-b)(a+b)) to each pair of terms. Look for a pattern in the resulting sum.

  8. Q8.hard

    Which of the following is the correct factorization of x⁴ + 4y⁴?
    1. A)(x² + 2y² - 2xy)(x² + 2y² + 2xy)
    2. B)(x² - 2y² - 2xy)(x² + 2y² + 2xy)
    3. C)(x² + 2y² - 4xy)(x² + 2y² + 4xy)
    4. D)(x² + 4y² - 2xy)(x² + 4y² + 2xy)
    Show answer

    Answer: (x² + 2y² - 2xy)(x² + 2y² + 2xy)

    Hint: This expression is not directly a difference of squares. Try adding and subtracting a term to create a perfect square, which then allows for the difference of squares identity.

  9. Q9.hard

    Factorize the expression: (x² + 3x + 1)(x² + 3x + 5) + 4.
    1. A)(x² + 3x + 3)²
    2. B)(x² + 3x + 2)²
    3. C)(x² + 3x + 1)(x² + 3x + 9)
    4. D)(x² + 3x - 2)(x² + 3x + 2)
    Show answer

    Answer: (x² + 3x + 3)²

    Hint: Notice the repeated expression 'x² + 3x'. Use substitution to simplify the problem into a more familiar quadratic form.

These are 9 of the 60 questions available for Factorisation. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.