Grade 8 Factorisation of Algebraic Expressions

Concept: **Factorisation** is the process of writing an algebraic expression as a product of two or more expressions. These expressions are called the factors of the original expression.

Analogy with Numbers: Just like we factorize a number (e.g., 12 = 2 * 6 or 12 = 3 * 4), we can factorize algebraic expressions.

Example:

The expression 5x + 10 can be written as 5(x + 2). Here, 5 and (x + 2) are the factors of 5x + 10.

The expression x² - y² can be written as (x + y)(x - y). Here, (x + y) and (x - y) are the factors of x² - y².

Step 1: Identify the terms in the expression. The terms are 6a and 9b.

Step 2: Find the greatest common factor (GCF) of the coefficients (6 and 9).

Factors of 6: 1, 2, 3, 6

Factors of 9: 1, 3, 9

The GCF of 6 and 9 is 3.

Step 3: Check for common variables in all terms. In 6a + 9b, the variable 'a' is only in the first term, and 'b' is only in the second term. There are no common variables.

Step 4: The common factor is the GCF of the coefficients, which is 3.

Step 5: Write the expression as the product of the common factor and the remaining expression. Divide each term by the common factor.

6a / 3 = 2a

9b / 3 = 3b

So, 6a + 9b = 3(2a + 3b)

Result: The factorised form is 3(2a + 3b).

Step 1: Identify the terms: 10x²y and -15xy².

Step 2: Find the GCF of the coefficients (10 and 15). The GCF is 5.

Step 3: Find the common variables with the lowest power.

Common variable 'x': x² and x¹. The lowest power is 1 (x).

Common variable 'y': y¹ and y². The lowest power is 1 (y).

The common variable factor is xy.

Step 4: The common factor is the product of the GCF of coefficients and the common variable factor: 5xy.

Step 5: Divide each term by the common factor.

10x²y / (5xy) = (10/5) * (x²/x) * (y/y) = 2 * x¹ * 1 = 2x

-15xy² / (5xy) = (-15/5) * (x/x) * (y²/y) = -3 * 1 * y¹ = -3y

So, 10x²y - 15xy² = 5xy(2x - 3y)

Result: The factorised form is 5xy(2x - 3y).

Concept: The difference of squares formula is: a² - b² = (a + b)(a - b).

Step 1: Recognize the expression as a difference of two perfect squares.

x² is the square of x (a = x).

25 is the square of 5 (b = 5).

The expression is in the form x² - 5².

Step 2: Apply the difference of squares formula: a² - b² = (a + b)(a - b).

Substitute a = x and b = 5.

x² - 5² = (x + 5)(x - 5)

Result: The factorised form is (x + 5)(x - 5).

Step 1: Recognize the expression as a difference of two perfect squares.

4p² is the square of 2p ((2p)² = 4p²). So, a = 2p.

49q² is the square of 7q ((7q)² = 49q²). So, b = 7q.

The expression is in the form (2p)² - (7q)².

Step 2: Apply the difference of squares formula: a² - b² = (a + b)(a - b).

Substitute a = 2p and b = 7q.

(2p)² - (7q)² = (2p + 7q)(2p - 7q)

Result: The factorised form is (2p + 7q)(2p - 7q).