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Mathematics · Class 9 · Chapter 4: Factorization

8 min

Key points

Factor — A factor of an algebraic expression is an expression which divides it exactly
Factorization — Factorization is the process of expressing an algebraic expression as a product of its factors
Common factor method — Take out the highest common factor (HCF) from all terms
Grouping method — Group terms with common factors and factorize each group separately
Using algebraic identities — Apply standard identities like a² - b², a² ± 2ab + b², a³ ± b³
Difference of two squares: a² - b² = (a + b)(a - b)
Perfect square trinomials: a² + 2ab + b² = (a + b)² and a² - 2ab + b² = (a - b)²
Sum and difference of cubes: a³ + b³ = (a + b)(a² - ab + b²) and a³ - b³ = (a - b)(a² + ab + b²)
Quadratic trinomials of the form ax² + bx + c can be factorized by splitting the middle term
Always check factorization by expanding the factors back to original expression

Definitions

Factor: A factor of an algebraic expression is an expression which divides it exactly.
Factorization: Factorization is the process of expressing an algebraic expression as a product of its factors.
Prime Factor: A prime factor is a factor that cannot be factorized further.
Common Factor: A common factor is a factor which is common to all terms of an algebraic expression.
Highest Common Factor (HCF): The highest common factor of two or more algebraic expressions is the expression of highest degree which divides each of them exactly.

Worked example

Factorize: 6x² + 11x + 3 (4 marks)

The factorized form is (2x + 3)(3x + 1)

Common mistakes

Students forget to check if there's a common factor before applying identities
Confusing a² + 2ab + b² = (a + b)² with a² - 2ab + b² = (a - b)²
Not verifying the factorization by expanding back to original expression
Making sign errors when applying difference of squares formula
Forgetting to factorize completely when multiple methods are needed

Quick recap

Factorization is expressing an algebraic expression as a product of its factors. Main methods include: taking out common factors, grouping method, and using special products (difference of squares, perfect square trinomials). Always verify your answer by expanding the factored form. Choose the appropriate method based on the structure of the given expression.

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