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About Factorization — Class 9 ICSE

Factorize polynomials using remainder theorem, factor theorem, and algebraic identities. This topic is part of the ICSE Class 9 mathematics syllabus (chapter: Chapter 4). On this page you can practice 59 questions across three difficulty levels — 20 easy, 20 medium, and 19 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: Unlocking Polynomials
  • Factorization Using Algebraic Identities: Patterns for Efficiency
  • Factorization of Quadratic Trinomials: Splitting the Middle Term
  • Remainder and Factor Theorems: Predicting Factors
  • Advanced Factorization and Exam Preparation

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

Factorization — solved examples for Class 9 ICSE

Example 1easy

Which of the following statements correctly describes the Remainder Theorem for a polynomial P(x)?
  1. A)If P(a) = 0, then (x - a) is a factor of P(x).
  2. B)When P(x) is divided by (x - a), the remainder is P(a).
  3. C)When P(x) is divided by (x + a), the remainder is P(a).
  4. D)If (x - a) is a factor of P(x), then P(a) ≠ 0.

Step-by-step solution

  1. The Remainder Theorem states that if a polynomial P(x) is divided by a linear polynomial (x - a), then the remainder is P(a).
  2. Option A describes the Factor Theorem. Option C has a sign error for the divisor (x + a). Option D contradicts the Factor Theorem.

Answer: When P(x) is divided by (x - a), the remainder is P(a).

Example 2medium

Factorize the expression x⁴ + 4 completely.
  1. A)(x² + 2)(x² + 2)
  2. B)(x² + 2x + 2)(x² - 2x + 2)
  3. C)(x² + 4)(x² + 1)
  4. D)(x² + 2x - 2)(x² - 2x - 2)

Step-by-step solution

  1. We can rewrite x⁴ + 4 as (x²)² + 2². To apply the identity a² - b², we need a middle term. Add and subtract 4x² (which is 2 × x² × 2).
  2. x⁴ + 4 = x⁴ + 4x² + 4 - 4x²
  3. Group the first three terms to form a perfect square: (x² + 2)² - (2x)².
  4. Now, apply the identity a² - b² = (a - b)(a + b), where a = (x² + 2) and b = 2x. This gives (x² + 2 - 2x)(x² + 2 + 2x), which can be written as (x² - 2x + 2)(x² + 2x + 2).

Answer: (x² + 2x + 2)(x² - 2x + 2)

Example 3hard

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

Step-by-step solution

  1. The expression `x⁴ + 4y⁴` can be rewritten as `(x²)² + (2y²)²`.
  2. To complete the square, we need to add and subtract `2(x²)(2y²) = 4x²y²`. So, `x⁴ + 4y⁴ = x⁴ + 4y⁴ + 4x²y² - 4x²y²`.
  3. This simplifies to `(x² + 2y²)² - (2xy)²`.
  4. Using the identity `a² - b² = (a - b)(a + b)`, where `a = (x² + 2y²)` and `b = 2xy`, we get `(x² + 2y² - 2xy)(x² + 2y² + 2xy)`.

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

Practice questions on Factorization

  1. Q1.easy

    Rohan attempted to factorize the polynomial 6x³ + 9x² + 4x + 6 using the grouping method. His steps are shown below:
    Step 1: 3x²(2x + 3) + 2(2x + 3)
    Step 2: (3x² + 2)(2x + 3)
    Which of the following statements identifies the error in Rohan's factorization, if any?
    1. A)There is no error; Rohan's factorization is correct.
    2. B)In Step 1, the common factor taken from the first two terms is incorrect.
    3. C)In Step 1, the common factor taken from the last two terms is incorrect.
    4. D)The polynomial cannot be factorized by grouping.
    Show answer

    Answer: There is no error; Rohan's factorization is correct.

    Hint: Expand Rohan's final factored form to check if it returns the original polynomial. If it does, then his factorization is correct.

  2. Q2.easy

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

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

    Hint: Recognize this expression as a difference of squares, A² - B², where A = (2x + 3y) and B = (x - y).

  3. Q3.easy

    For a polynomial P(x), if P(-2) = 0, which of the following statements must be true according to the Factor Theorem?
    1. A)(x + 2) is a factor of P(x).
    2. B)(x - 2) is a factor of P(x).
    3. C)The remainder when P(x) is divided by (x - 2) is 0.
    4. D)P(x) has no real factors.
    Show answer

    Answer: (x + 2) is a factor of P(x).

    Hint: The Factor Theorem states that if P(a) = 0, then (x - a) is a factor of P(x). Pay close attention to the sign.

  4. Q4.medium

    Factorize the expression a² - b² - 2a + 1 completely.
    1. A)(a - b - 1)(a + b - 1)
    2. B)(a + b - 1)(a - b + 1)
    3. C)(a - 1 - b)(a - 1 + b)
    4. D)(a + 1 - b)(a - 1 + b)
    Show answer

    Answer: (a - 1 - b)(a - 1 + b)

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

  5. Q5.medium

    If (x - 2) and (x + 3) are factors of the polynomial P(x) = x³ + ax² + bx - 12, find the values of a and b.
    1. A)a = 3, b = -4
    2. B)a = -1, b = 8
    3. C)a = 2, b = -6
    4. D)a = -2, b = 6
    Show answer

    Answer: a = 3, b = -4

    Hint: Use the Factor Theorem: if (x - c) is a factor of P(x), then P(c) = 0. This will give you a system of linear equations.

  6. Q6.medium

    When the polynomial P(x) = ax³ + 3x² + bx - 13 is divided by (x - 2), the remainder is 5. When it is divided by (x + 1), the remainder is -10. Find the values of a and b.
    1. A)a = 2, b = -7
    2. B)a = -1, b = 5
    3. C)a = 3, b = -9
    4. D)a = 1, b = -1
    Show answer

    Answer: a = 1, b = -1

    Hint: Apply the Remainder Theorem: if P(x) is divided by (x - c), the remainder is P(c). Set up two equations.

  7. Q7.hard

    Factorize the expression `x² - (a + 1/a)x + 1`.
    1. A)(x - a)(x - 1/a)
    2. B)(x + a)(x - 1/a)
    3. C)(x - a)(x + 1/a)
    4. D)(x + a)(x + 1/a)
    Show answer

    Answer: (x - a)(x - 1/a)

    Hint: This is a quadratic trinomial. Try to split the middle term `-(a + 1/a)x` into two parts whose product is `1 × 1 = 1`.

  8. Q8.hard

    If `(x - 1)` and `(x + 2)` are factors of the polynomial `P(x) = x³ + ax² - bx - 2`, what are the values of `a` and `b` respectively?
    1. A)a = -1, b = -2
    2. B)a = 1, b = 2
    3. C)a = 2, b = 1
    4. D)a = -2, b = -1
    Show answer

    Answer: a = 2, b = 1

    Hint: Apply the Factor Theorem for both given factors. This will give you a system of two linear equations in `a` and `b`.

  9. Q9.hard

    The polynomial `P(x) = x³ + kx² + x + 2` leaves the same remainder when divided by `(x - 1)` and `(x + 2)`. What is the value of `k`?
    1. A)k = 1
    2. B)k = 2
    3. C)k = 3
    4. D)k = 4
    Show answer

    Answer: k = 4

    Hint: According to the Remainder Theorem, the remainder when `P(x)` is divided by `(x - a)` is `P(a)`. Set `P(1)` equal to `P(-2)`.

These are 9 of the 59 questions available for Factorization. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.