Factorisation of Algebraic Expressions and Identities for Class 8 Maths

What is Factorisation of Algebraic Expressions?

In mathematics, an algebraic expression is a combination of variables (like x, y, a, b), constants (like 2, 5, -7), and arithmetic operations (+, -, x, /). For example, 5x + 3, x^2 - 4y, and 2a^2 + 3ab + b^2 are all algebraic expressions.

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. Think of it like this: when we say that 12 = 2 \times 6, 2 and 6 are factors of 12. Similarly, if we have an expression 2x + 4, we can write it as 2(x + 2). Here, 2 and (x + 2) are the factors of 2x + 4.

This process is incredibly useful because it helps us to:

1. Simplify expressions: A factorised expression is often easier to work with.
2. Solve equations: Factorisation is a key step in solving quadratic equations and other polynomial equations.
3. Understand relationships: It reveals the fundamental building blocks of an expression.

We will explore different techniques to factorise expressions, starting with the simplest ones and moving towards more complex methods involving special algebraic identities.

Key Methods of Factorisation

1. Method 1: Taking Out Common Factors — This is the most basic method. We look for a factor that is present in every term of the expression. Steps: 1. Identify the terms: Separate the expression into individual terms. 2. Find the Greatest Common Factor (GCF): Determine the largest number and the highest power of variables that divides all terms. 3. Divide each term by the GCF: Write the GCF outside parentheses and the result of the division inside. Example: Factorise 6x^2y - 9xy^2 Terms are 6x^2y and -9xy^2. GCF of 6 and 9 is 3. GCF of x^2 and x is x. GCF of y and y^2 is y. So, the overall GCF is 3xy. Divide 6x^2y by 3xy to get 2x. Divide -9xy^2 by 3xy to get -3y. Therefore, 6x^2y - 9xy^2 = 3xy(2x - 3y).
2. Method 2: Factorisation by Regrouping Terms — Sometimes, there isn't a common factor for all terms, but groups of terms might have common factors. In such cases, we regroup the terms. Steps: 1. Arrange and Regroup: Look for terms that share common factors and group them together. You might need to rearrange the terms. 2. Factorise Each Group: Take out the common factor from each group. 3. Identify Common Binomial Factor: After factoring each group, you should find a common binomial (an expression with two terms) factor. 4. Factor Out the Common Binomial: Take out the common binomial as a single factor. Example: Factorise ax + bx + ay + by Regroup: (ax + bx) + (ay + by) Factor each group: x(a + b) + y(a + b) Identify common binomial: (a + b) is common. Factor out: (a + b)(x + y).

Factorisation Using Algebraic Identities

• Identity 1: a^2 + 2ab + b^2 = (a+b)^2
  This identity is used when you have a three-term expression where the first and last terms are perfect squares and the middle term is twice the product of their square roots.

  Example: Factorise x^2 + 10x + 25
  Step 1: Check if the first and last terms are perfect squares. Here, x^2 = (x)^2 and 25 = (5)^2.
  Step 2: Check if the middle term is 2ab. Here, 2 \times x \times 5 = 10x. This matches the middle term.
  Step 3: Apply the identity. Since a=x and b=5, the factorised form is (x+5)^2.

• Identity 2: a^2 - 2ab + b^2 = (a-b)^2
  Similar to the previous identity, but used when the middle term is negative.

  Example: Factorise 9y^2 - 12y + 4
  Step 1: Check if the first and last terms are perfect squares. Here, 9y^2 = (3y)^2 and 4 = (2)^2.
  Step 2: Check if the middle term is -2ab. Here, -2 \times 3y \times 2 = -12y. This matches the middle term.
  Step 3: Apply the identity. Since a=3y and b=2, the factorised form is (3y-2)^2.

• Identity 3: a^2 - b^2 = (a-b)(a+b)
  This identity is used for expressions with two terms, where both terms are perfect squares and are separated by a minus sign.

  Example: Factorise 49p^2 - 81q^2
  Step 1: Check if both terms are perfect squares. Here, 49p^2 = (7p)^2 and 81q^2 = (9q)^2.
  Step 2: Identify 'a' and 'b'. Here, a = 7p and b = 9q.
  Step 3: Apply the identity. The factorised form is (7p - 9q)(7p + 9q).

Important Tips for Factorisation

Factorisation can sometimes feel tricky, but with a systematic approach, you can master it. Here are some essential tips for your CBSE Class 8 exams:

1. Always Look for Common Factors First: Before trying any other method (like regrouping or identities), always check if there"s a common factor among all terms. Taking out the GCF simplifies the expression and often makes other methods easier to spot.
2. Recognise Patterns for Identities: Practice identifying expressions that fit the a^2 + 2ab + b^2, a^2 - 2ab + b^2, or a^2 - b^2 forms. Look for perfect squares at the beginning and end (or two terms that are perfect squares with a minus sign in between).
3. Check Your Answer: After you factorise an expression, always multiply the factors back. If you get the original expression, your factorisation is correct! For example, if you factorised 6x + 9 as 3(2x + 3), multiply 3 \times 2x and 3 \times 3 to get 6x + 9.
4. Be Careful with Signs: A single sign error can lead to incorrect factorisation, especially with identities like (a-b)^2 or when factoring out negative common terms.
5. Try Different Methods: If one method doesn't seem to work, don't get stuck. Try another. For instance, if common factors aren't apparent, consider regrouping or looking for identity patterns.

Practice Questions with Solutions

• Q: Factorise: 14pq + 35p
  A: Step 1: Identify the terms: 14pq and 35p.
  Step 2: Find the GCF of 14 and 35, which is 7. Find the GCF of pq and p, which is p.
  Step 3: The overall GCF is 7p.
  Step 4: Divide each term by 7p: 14pq / 7p = 2q and 35p / 7p = 5.
  Final answer: 7p(2q + 5)

• Q: Factorise: xy + 2x + 3y + 6
  A: Step 1: Group terms: (xy + 2x) + (3y + 6).
  Step 2: Factor out common factors from each group: x(y + 2) + 3(y + 2).
  Step 3: Notice the common binomial factor (y + 2).
  Step 4: Factor out the common binomial.
  Final answer: (y + 2)(x + 3)

• Q: Factorise: 4m^2 + 20m + 25
  A: Step 1: Check if the first and last terms are perfect squares: 4m^2 = (2m)^2 and 25 = (5)^2.
  Step 2: Check if the middle term is 2ab: 2 \times (2m) \times (5) = 20m.
  Step 3: This matches the identity a^2 + 2ab + b^2 = (a+b)^2 where a=2m and b=5.
  Final answer: (2m + 5)^2

• Q: Factorise: 81x^2 - 1
  A: Step 1: Check if both terms are perfect squares and separated by a minus sign: 81x^2 = (9x)^2 and 1 = (1)^2.
  Step 2: This matches the identity a^2 - b^2 = (a-b)(a+b) where a=9x and b=1.
  Final answer: (9x - 1)(9x + 1)

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