Factorisation Class 8 Maths Notes | YoLearn.ai

What is Factorisation?

In mathematics, specifically in algebra, factorisation is the process of breaking down an algebraic expression or a number into a product of its factors. Think of it as the reverse operation of multiplication. When you multiply two or more expressions, you get a product. Factorisation takes that product and finds the expressions that were multiplied together to get it. For example, multiplying (x + 2) by (x + 3) gives x² + 5x + 6. Factorisation of x² + 5x + 6 would yield (x + 2)(x + 3).

The primary goal of factorisation is to simplify expressions, solve equations, and understand the structure of polynomials. An expression is said to be factorised completely when it cannot be broken down further into simpler algebraic factors. These factors are often called irreducible factors. It's a fundamental skill that underpins much of advanced algebra and calculus, making its mastery in Class 8 essential.

Key Definitions

Factor (of an algebraic expression)

An expression that divides another expression exactly, leaving no remainder. When multiplied with other factors, it gives the original expression.

Irreducible Factor

A factor that cannot be expressed as a product of factors of lower degree or simpler forms. For example, (x+y) is an irreducible factor.

Common Factor

A factor that is present in all terms of an algebraic expression.

Algebraic Expression

A combination of constants, variables, and algebraic operations (addition, subtraction, multiplication, division).

Polynomial

An algebraic expression consisting of one or more terms, each of which is a product of a constant and one or more variables raised to non-negative integer powers.

Identity

An equation that is true for all values of the variables involved. Algebraic identities are often used as formulas for factorisation.

Methods of Factorisation

1. 1. Method of Common Factors —
2. 2. Factorisation by Grouping Terms —
3. 3. Factorisation using Algebraic Identities —
4. 4. Factorisation of Polynomials of the form x² + bx + c (Splitting the Middle Term) —

Solved Examples

• {"title":"Example 1: Factor by Common Factor","description":"Factorise 5x²y - 15xy²","steps":["5x²y - 15xy²","Identify common factors: numerical 5, variable x and y.","GCF = 5xy","5xy(x - 3y)"]}
• {"title":"Example 2: Factor using Identity","description":"Factorise 25p² - 40pq + 16q²","steps":["25p² - 40pq + 16q²","Recognise the form a² - 2ab + b².","Here, a = √(25p²) = 5p and b = √(16q²) = 4q.","Check middle term: 2ab = 2(5p)(4q) = 40pq. This matches.","So, it is (5p - 4q)²."]}
• {"title":"Example 3: Factor by Splitting the Middle Term","description":"Factorise m² - 7m - 18","steps":["m² - 7m - 18","Find two numbers whose product is -18 and sum is -7.","Numbers are -9 and 2 (-9 × 2 = -18, -9 + 2 = -7).","Rewrite: m² - 9m + 2m - 18","Group terms: m(m - 9) + 2(m - 9)","Factor out (m - 9): (m - 9)(m + 2)"]}

Key Points to Remember

• Factorisation is the reverse of multiplication/expansion.
• An expression is completely factorised when all its factors are irreducible.
• Always look for a common factor first, as it simplifies the expression significantly.
• Memorise the three basic algebraic identities: (a+b)², (a-b)², a²-b² for quick application.
• Splitting the middle term method is crucial for quadratic trinomials of the form x² + bx + c or ax² + bx + c.
• Check your factorisation by multiplying the factors back to see if you get the original expression.
• For factorisation by grouping, the common binomial factor must be exactly the same in each grouped term.

Exam Tip: Avoiding Common Traps

Many students lose marks by not factorising completely. Always check if the resulting factors can be further broken down. For instance, x⁴ - y⁴ is (x² - y²)(x² + y²) but (x² - y²) can be factorised further to (x-y)(x+y). So, the complete factorisation is (x-y)(x+y)(x²+y²). Also, be careful with signs when splitting the middle term; a small sign error can lead to incorrect factors. Practice applying the algebraic identities in reverse – this is key to mastering factorisation questions that use these patterns. Double-check your signs, especially in identities like a² - b².

Practice Questions with Solutions

• Q: What is an irreducible factor? A: An irreducible factor is a factor that cannot be broken down further into simpler algebraic factors of lower degree.
• Q: Factorise 6a²b + 9ab². A: The common factor is 3ab. So, 3ab(2a + 3b).
• Q: Which identity would you use to factorise 49m² - 64n²? A: The identity a² - b² = (a - b)(a + b).
• Q: What two numbers would you use to split the middle term for y² - 10y + 21? A: The two numbers are -3 and -7, because (-3) + (-7) = -10 and (-3) × (-7) = 21.

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