What is Factorisation Using Identities? | Simple Explanation for Class 6

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What is the Factorisation Using Identities?

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

Factorisation using identities is a special way to break down a mathematical expression into simpler parts (factors) by recognising common patterns. We use algebraic identities, which are like special shortcut formulas, to make this process quicker and easier. It helps us rewrite complex expressions as a product of simpler ones.

Simple Example

Quick Example

Imagine you have a big square mithai box where the lid's area is given by the expression 'x^2 + 6x + 9'. If you recognise this as the pattern of (a+b)^2, you can quickly say the side length of the lid is '(x+3)'. So, 'x^2 + 6x + 9' is factorised into '(x+3)(x+3)'.

Worked Example

Step-by-Step

Let's factorise the expression: 4x^2 + 12x + 9

1. Look for a pattern that matches one of the algebraic identities. The expression 4x^2 + 12x + 9 looks like (a+b)^2 = a^2 + 2ab + b^2.
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2. Identify 'a' and 'b'. Here, a^2 = 4x^2, so a = sqrt(4x^2) = 2x.
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3. Similarly, b^2 = 9, so b = sqrt(9) = 3.
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4. Now, check if the middle term, 2ab, matches the middle term of our expression. 2ab = 2 * (2x) * (3) = 12x.
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5. Since 12x matches the middle term in our original expression, we can confirm it fits the (a+b)^2 identity.
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6. Replace 'a' and 'b' in the identity (a+b)^2. So, (2x + 3)^2.
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Answer: The factorisation of 4x^2 + 12x + 9 is (2x + 3)(2x + 3).

Why It Matters

Factorisation is super important for solving complex problems in science and technology. Engineers use it to design bridges and buildings, while computer scientists use it in coding and developing secure systems. Understanding this helps you think logically, which is key for careers in AI, Data Science, and even making better economic predictions.

Common Mistakes

MISTAKE: Not checking the middle term (2ab) when using (a+b)^2 or (a-b)^2. Students often only look at the first and last terms. | CORRECTION: Always verify that the middle term of the expression exactly matches 2ab (or -2ab) after finding 'a' and 'b' from the squared terms.

MISTAKE: Confusing (a-b)^2 with a^2 - b^2. For example, factorising x^2 - 9 as (x-3)^2. | CORRECTION: Remember (a-b)^2 = a^2 - 2ab + b^2, which has three terms. The identity a^2 - b^2 = (a-b)(a+b) has only two terms, separated by a minus sign.

MISTAKE: Incorrectly finding 'a' or 'b' from the squared terms, especially with coefficients. For example, saying 'a' is 2x^2 for 4x^2. | CORRECTION: 'a' and 'b' are the square roots of the terms. For 4x^2, a = sqrt(4x^2) = 2x, not 2x^2.

Practice Questions

Try It Yourself

QUESTION: Factorise: x^2 + 10x + 25 | ANSWER: (x + 5)(x + 5)

QUESTION: Factorise: 9y^2 - 30y + 25 | ANSWER: (3y - 5)(3y - 5)

QUESTION: Factorise: 16p^2 - 49q^2 | ANSWER: (4p - 7q)(4p + 7q)

MCQ

Quick Quiz

Which identity would you use to factorise x^2 - 81?

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

a^2 - b^2 = (a - b)(a + b)

(x + a)(x + b) = x^2 + (a+b)x + ab

The Correct Answer Is:

The expression x^2 - 81 has two terms, both are perfect squares, and they are separated by a minus sign. This perfectly matches the form a^2 - b^2 = (a - b)(a + b).

Real World Connection

In the Real World

Imagine you're a cricket analyst trying to predict player performance. Sometimes, complex statistical formulas can be simplified using factorisation. This helps you quickly understand patterns in runs scored or wickets taken, giving your team an edge in strategising for the next match, just like how companies like Dream11 use data analytics.

Key Vocabulary

Key Terms

FACTORISATION: Breaking down an expression into simpler parts (factors) whose product is the original expression. | IDENTITY: An equation that is true for all possible values of its variables. | ALGEBRAIC EXPRESSION: A combination of variables, numbers, and arithmetic operations. | TERM: A single number or variable, or numbers and variables multiplied together, separated by + or - signs. | SQUARE ROOT: A number that, when multiplied by itself, gives the original number.

What's Next

What to Learn Next

Great job learning about factorisation using basic identities! Next, you can explore 'Factorisation by Splitting the Middle Term' and 'Factorisation by Grouping'. These methods will give you more tools to factorise even more complex expressions, making you a math wizard!