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About Factorization — Class 8 ICSE

Factorize algebraic expressions using common factors, regrouping, and identities. This topic is part of the ICSE Class 8 mathematics syllabus (chapter: Chapter 6). 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.

What you'll learn in Factorization

  • Introduction to Factorization: The Reverse of Multiplication
  • Factorization by Taking Out Common Monomial Factors
  • Factorization by Regrouping Terms
  • Factorization Using Algebraic Identities
  • Comprehensive Factorization: Combining Methods

Interactive lesson · about 15 minutes · checkpoint question after every unit

Factorization — solved examples for Class 8 ICSE

Example 1easy

Which of the following expressions has 3xy as its Highest Common Factor (HCF) among its terms?
  1. A)6x²y + 9xy²
  2. B)3x²y + 6xy
  3. C)9x²y² + 12xy
  4. D)3xy + 6x²y²

Step-by-step solution

  1. For option A) 6x²y + 9xy²:
  2. The HCF of the coefficients (6 and 9) is 3.
  3. The HCF of the variable parts (x²y and xy²) is xy (lowest power of x is x¹, lowest power of y is y¹).
  4. Therefore, the HCF of 6x²y and 9xy² is 3xy.

Answer: 6x²y + 9xy²

Example 2medium

Factorize the expression: 12x³y² - 18x²y³ + 24x⁴y
  1. A)6x²y(2xy - 3y² + 4x²)
  2. B)6x²y(2xy - 3y² + 4x³)
  3. C)6x²y(2xy - 3y² + 4x)
  4. D)6xy(2x²y - 3xy² + 4x³)

Step-by-step solution

  1. Identify the greatest common factor (GCF) of the numerical coefficients (12, 18, 24). The GCF is 6.
  2. Identify the lowest power of each variable common to all terms. For 'x', it's x²; for 'y', it's y. So the common variable factor is x²y.
  3. The overall GCF is 6x²y. Divide each term by this GCF:
  4. 12x³y² / 6x²y = 2xy
    12x3y2−18x2y3+24x4y=6x2y(2xy−3y2+4x2)12x³y² - 18x²y³ + 24x⁴y = 6x²y(2xy - 3y² + 4x²)
  5. -18x²y³ / 6x²y = -3y²
  6. 24x⁴y / 6x²y = 4x²
  7. Combine these to get the factored form: 6x²y(2xy - 3y² + 4x²).

Answer: 6x²y(2xy - 3y² + 4x²)

Example 3hard

Factorize completely: 0.25a³b² - 0.5a²b³ + 0.75ab⁴
  1. A)0.25ab(a²b - 2ab² + 3b³)
  2. B)0.25ab²(a² - 2ab + 3b²)
  3. C)0.25ab²(a² - 2b + 3b²)
  4. D)0.25ab²(a² - 2ab + 3b)

Step-by-step solution

  1. Identify the GCF of the numerical coefficients (0.25, 0.5, 0.75). The GCF is 0.25.
  2. Identify the GCF of the variable terms (a³b², a²b³, ab⁴). The GCF is ab².
  3. Combine these to get the overall GCF: 0.25ab².
  4. Divide each term in the expression by the GCF and write the remaining terms inside the parenthesis: 0.25ab²(a³b²/0.25ab² - 0.5a²b³/0.25ab² + 0.75ab⁴/0.25ab²) = 0.25ab²(a² - 2ab + 3b²).

Answer: 0.25ab²(a² - 2ab + 3b²)

Practice questions on Factorization

  1. Q1.easy

    Rohan is trying to factorize the expression 2ax + 3by + 2bx + 3ay. He rearranges the terms as (2ax + 2bx) + (3by + 3ay). Which factorization method is Rohan applying?
    1. A)Taking out common monomial factor
    2. B)Grouping terms
    3. C)Using identity a² - b²
    4. D)Splitting the middle term
    Show answer

    Answer: Grouping terms

    Hint: Observe how Rohan combined terms to find common factors within those specific groups, even if there isn't a common factor for the entire expression initially.

  2. Q2.easy

    A student tried to factorize 5x(p-q) - 7y(q-p). They wrote 5x(p-q) - 7y(p-q) = (p-q)(5x - 7y). Is this factorization correct? If not, what was the mistake?
    1. A)Correct
    2. B)Incorrect, the common factor is (q-p)
    3. C)Incorrect, the final expression should be (p-q)(5x + 7y)
    4. D)Incorrect, the second term should be -7y(p-q)
    Show answer

    Answer: Incorrect, the final expression should be (p-q)(5x + 7y)

    Hint: Recall the relationship between (p-q) and (q-p). How does a negative sign affect the terms inside a bracket?

  3. Q3.easy

    Which of the following statements about factorization is TRUE?
    1. A)Factorization breaks an expression into a sum of simpler terms.
    2. B)An expression is completely factorized when it's written as a product of factors that cannot be factorized further.
    3. C)All algebraic expressions can be factorized using identities.
    4. D)Taking out a common factor requires all terms to have identical variable parts.
    Show answer

    Answer: An expression is completely factorized when it's written as a product of factors that cannot be factorized further.

    Hint: Think about the primary goal and definition of factorization. What is the final form we aim for?

  4. Q4.medium

    Factorize: 3ax - 6ay - 8by + 4bx
    1. A)(3a + 4b)(x - 2y)
    2. B)(3a - 4b)(x + 2y)
    3. C)(3a + 4b)(x + 2y)
    4. D)(3a - 4b)(x - 2y)
    Show answer

    Answer: (3a + 4b)(x - 2y)

    Hint: Rearrange the terms if necessary to find common factors within groups, then look for a common binomial factor.

  5. Q5.medium

    Factorize: 49p² - 81q²
    1. A)(7p - 9q)²
    2. B)(7p + 9q)²
    3. C)(7p - 9q)(7p + 9q)
    4. D)(49p - 81q)(49p + 81q)
    Show answer

    Answer: (7p - 9q)(7p + 9q)

    Hint: Recognize the pattern of the difference of two squares, a² - b² = (a - b)(a + b).

  6. Q6.medium

    Factorize: 25x² + 60xy + 36y²
    1. A)(5x - 6y)²
    2. B)(5x + 6y)²
    3. C)(5x + 6y)(5x - 6y)
    4. D)(25x + 36y)²
    Show answer

    Answer: (5x + 6y)²

    Hint: Check if the first and last terms are perfect squares, and if the middle term is twice the product of their square roots.

  7. Q7.hard

    Factorize: x² - y² - 2x + 1
    1. A)(x - y - 1)(x + y - 1)
    2. B)(x + 1 - y)(x + 1 + y)
    3. C)(x - 1 - y)(x - 1 + y)
    4. D)(x - y + 1)(x + y - 1)
    Show answer

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

    Hint: Look for a perfect square trinomial hidden within the first three terms. Once identified, you can apply the difference of two squares identity.

  8. Q8.hard

    Factorize: (2a - b)² - (a - 2b)²
    1. A)3(a - b)(a + 3b)
    2. B)3(a - b)(a + b)
    3. C)3(a + b)(a + b)
    4. D)3(a - b)(a - b)
    Show answer

    Answer: 3(a - b)(a + b)

    Hint: This expression is in the form of a² - b². Directly apply the difference of two squares identity, treating each binomial as a single term.

  9. Q9.hard

    Which of the following expressions is NOT a perfect square trinomial?
    1. A)4x² + 12xy + 9y²
    2. B)x⁴ - 10x² + 25
    3. C)9a² - 30ab + 25b²
    4. D)16m² + 20mn + 25n²
    Show answer

    Answer: 16m² + 20mn + 25n²

    Hint: Recall the conditions for a perfect square trinomial: a² + 2ab + b² or a² - 2ab + b². The middle term must be exactly twice the product of the square roots of the first and last terms.

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