Factorisation Chapter Notes | Mathematics Class 9 ICSE PDF Download

Introduction

Factorisation is like solving a puzzle where you break down a complex expression into simpler pieces that multiply together to give the original expression. Imagine it as taking apart a toy to see its building blocks! This chapter introduces you to the art of expressing polynomials as products of their factors, making it easier to solve equations, simplify expressions, and understand algebraic structures. By learning different methods of factorisation, you'll unlock the ability to tackle a wide range of mathematical challenges with confidence.

• Factorisation is the process of writing a polynomial as a product of two or more simpler expressions, called factors.
• Each factor is a term or expression that, when multiplied, gives back the original polynomial.
• It is the reverse of multiplication, breaking down expressions into their building blocks.

Example: The polynomial x² + 5x + 6 can be written as (x + 3)(x + 2).
Here, (x + 3) and (x + 2) are factors, because (x + 3)(x + 2) = x² + 5x + 6.

Methods of Factorisation

Type 1: Taking out the Common Factors

• Identify the highest common factor (HCF) present in all terms of the expression.
• Divide each term by this HCF and write the quotient inside brackets, with the HCF outside.
• This method simplifies the expression by factoring out the common term.

Example: For 8ab² + 12a²b:

• Step 1: Find the HCF of 8ab² and 12a²b, which is 4ab.
• Step 2: Divide each term by 4ab: (8ab² ÷ 4ab) = 2b and (12a²b ÷ 4ab) = 3a.
• Step 3: Write as 4ab(2b + 3a).

Type 2: Grouping

• This method is used for expressions with an even number of terms, where terms can be grouped to have common factors.
• Group the terms in pairs or sets so that each group shares a common factor.
• Factorise each group separately by taking out the common factor.
• Take the common factor from the resulting groups to complete the factorisation.

Example: For ab + bc + ax + cx:

• Step 1: Group terms: (ab + bc) + (ax + cx).
• Step 2: Factorise each group: b(a + c) from the first group, x(a + c) from the second.
• Step 3: Take out the common factor (a + c): (a + c)(b + x).

Type 3: Trinomial of the Form ax2 ± bx ± c (By Splitting the Middle Term)

• This method applies to quadratic expressions of the form ax² + bx + c.
• Split the middle term (bx) into two parts whose sum is b and product is a × c.
• Group the terms and factorise by taking common factors.

Example: For x² + 5x + 6:

• Step 1: Identify a = 1, b = 5, c = 6. Find two numbers whose sum is 5 and product is 1 × 6 = 6 (numbers are 3 and 2).
• Step 2: Rewrite as x² + 3x + 2x + 6.
• Step 3: Group: x(x + 3) + 2(x + 3).
• Step 4: Factor out: (x + 3)(x + 2).

Type 4: Difference of Two Squares

• Expressions of the form x² - y² can be factorised as (x + y)(x - y).
• Rewrite the expression to identify it as a difference of squares, then apply the formula.

Example: For x² - 25:

• Step 1: Recognize 25 as 5², so x² - 25 = x² - 5².
• Step 2: Apply the formula: x² - 5² = (x + 5)(x - 5).

Type 5: The Sum or Difference of Two Cubes

• Sum of cubes: a³ + b³ = (a + b)(a² - ab + b²).
• Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²).
• Identify the expression as a sum or difference of cubes, then apply the appropriate formula.

Example: For a³ + 27b³:

• Step 1: Rewrite 27b³ as (3b)³, so a³ + 27b³ = a³ + (3b)³.
• Step 2: Apply sum of cubes formula: a³ + (3b)³ = (a + 3b)(a² - a·3b + (3b)²).
• Step 3: Simplify: (a + 3b)(a² - 3ab + 9b²).