Factorisation Exercise 14.1: Finding Common Factors (Class 8 Maths)

Understanding Factorisation and Common Factors

Factorisation is essentially the reverse process of multiplication in algebra. When we multiply expressions like 2x and (x + 3), we get 2x² + 6x. Factorisation involves starting with 2x² + 6x and breaking it down into 2x(x + 3). The terms 2x and (x + 3) are called the factors of 2x² + 6x.

The first and most basic method of factorisation is finding common factors. A common factor is a term that divides each term in the given algebraic expression exactly. For example, in the expression 3x + 9, both 3x and 9 can be divided by 3. So, 3 is a common factor. Similarly, in 4xy + 8x, both terms can be divided by 4 and by x, making 4x a common factor. Identifying and extracting these common factors simplifies expressions and helps in solving equations later on.

Step-by-Step: Finding Common Factors in Expressions

1. Step 1: Identify Each Term — Clearly separate the algebraic expression into individual terms. For example, in 12x + 18y, the terms are 12x and 18y. In 5x²y - 15xy² + 10xyz, the terms are 5x²y, -15xy², and 10xyz.
2. Step 2: Factorise Each Term Numerically — Break down the numerical coefficient of each term into its prime factors. For example, 12 = 2 × 2 × 3, 18 = 2 × 3 × 3.
3. Step 3: Factorise Each Term Algebraically — Write out all the variable factors for each term. Remember that x² means x × x, y³ means y × y × y, and so on. For 12x, this is 2 × 2 × 3 × x. For 18y, this is 2 × 3 × 3 × y.
4. Step 4: Identify Common Factors — Look for factors (both numerical and variable) that are present in ALL the factorised terms. If a factor appears multiple times in each term, take the lowest power of that factor. For 12x (2 × 2 × 3 × x) and 18y (2 × 3 × 3 × y), the common numerical factors are 2 and 3. So, 2 × 3 = 6 is the common numerical factor. There are no common variable factors here. The overall common factor is 6.
5. Step 5: Write the Factorised Expression — Write the common factor outside a bracket, and inside the bracket, write the remaining factors of each term. For 12x + 18y, the common factor is 6. 12x / 6 = 2x and 18y / 6 = 3y. So, the factorised expression is 6(2x + 3y).

Factorising by Taking Out Common Factors

• Example 1: Factorise 7x - 42 Step 1: Terms are 7x and -42. Step 2: Numerical factors: 7x = 7 × x, 42 = 2 × 3 × 7. Step 3: Common numerical factor is 7. There are no common variable factors. Step 4: Take 7 out as the common factor. Step 5: 7x / 7 = x, -42 / 7 = -6. Final Answer: 7(x - 6)
• Example 2: Factorise 6pq - 9qr Step 1: Terms are 6pq and -9qr. Step 2: Numerical factors: 6 = 2 × 3, 9 = 3 × 3. Step 3: Variable factors: pq = p × q, qr = q × r. Step 4: Common numerical factor is 3. Common variable factor is q. So, the common factor is 3q. Step 5: 6pq / 3q = 2p, -9qr / 3q = -3r. Final Answer: 3q(2p - 3r)
• Example 3: Factorise 10a² - 15b² + 20c² Step 1: Terms are 10a², -15b², 20c². Step 2: Numerical factors: 10 = 2 × 5, 15 = 3 × 5, 20 = 2 × 2 × 5. Step 3: Variable factors: a² = a × a, b² = b × b, c² = c × c. Step 4: Common numerical factor is 5. There are no common variable factors among all three terms. Step 5: 10a² / 5 = 2a², -15b² / 5 = -3b², 20c² / 5 = 4c². Final Answer: 5(2a² - 3b² + 4c²)

YoLearn AI Tutor Exam Tip: Double-Check Your Factors!

After you factorise an expression, always perform a quick mental check by multiplying the factors back. For example, if you factorised 6x + 10 as 2(3x + 5), mentally multiply 2 × 3x = 6x and 2 × 5 = 10. Adding them back gives 6x + 10, which is the original expression. If the result doesn't match, you've made a mistake! This simple step can save you marks in exams. Also, ensure you've taken out the greatest common factor (GCF). Sometimes students might take out a small common factor, but not the largest one, leaving the expression not fully factorised.

Practice Questions with Solutions

• Q: Factorise 12x + 36. A: Step 1: Identify terms: 12x and 36. Step 2: Find prime factors: 12 = 2 × 2 × 3, 36 = 2 × 2 × 3 × 3. Step 3: Common numerical factors are 2, 2, 3. So, GCF of numbers is 2 × 2 × 3 = 12. There are no common variable factors. Step 4: Divide each term by the GCF (12): 12x / 12 = x; 36 / 12 = 3. Final answer: 12(x + 3)
• Q: Factorise 14p - 21q. A: Step 1: Identify terms: 14p and -21q. Step 2: Find prime factors: 14 = 2 × 7, 21 = 3 × 7. Step 3: Common numerical factor is 7. There are no common variable factors. Step 4: Divide each term by the GCF (7): 14p / 7 = 2p; -21q / 7 = -3q. Final answer: 7(2p - 3q)
• Q: Factorise 20l²m + 30alm. A: Step 1: Identify terms: 20l²m and 30alm. Step 2: Find prime factors: 20 = 2 × 2 × 5, 30 = 2 × 3 × 5. Step 3: Identify common numerical factors (2, 5). GCF of numbers = 2 × 5 = 10. Identify common variable factors (l, m). GCF of variables = l × m = lm. Step 4: GCF for the expression is 10lm. Divide each term by 10lm: 20l²m / 10lm = 2l; 30alm / 10lm = 3a. Final answer: 10lm(2l + 3a)
• Q: Factorise 5x²y - 15xy². A: Step 1: Identify terms: 5x²y and -15xy². Step 2: Find prime factors: 5 = 5, 15 = 3 × 5. Step 3: Identify common numerical factor (5). GCF of numbers = 5. Identify common variable factors (x, y). GCF of variables = x × y = xy. (Note: x² has x and x, xy² has x and y and y. Common is one x and one y). Step 4: GCF for the expression is 5xy. Divide each term by 5xy: 5x²y / 5xy = x; -15xy² / 5xy = -3y. Final answer: 5xy(x - 3y)

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