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What is the Factorisation of Difference of Two Cubes?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The factorisation of the difference of two cubes is a special algebraic identity that helps break down an expression where one perfect cube is subtracted from another perfect cube. It states that a^3 - b^3 can be written as (a - b)(a^2 + ab + b^2). This formula helps simplify complex expressions into simpler factors.
Simple Example
Quick Example
Imagine you have two large boxes of ladoos. One box has 'a' ladoos in each dimension (a x a x a = a^3 total ladoos), and the other has 'b' ladoos in each dimension (b x b x b = b^3 total ladoos). If you want to find the difference in the number of ladoos, a^3 - b^3, this formula helps you find two simpler expressions (factors) that multiply to give you that difference.
Worked Example
Step-by-Step
Let's factorise x^3 - 8.
---Step 1: Identify 'a' and 'b'. Here, a^3 = x^3, so a = x. Also, b^3 = 8. We know that 2 x 2 x 2 = 8, so b = 2.
---Step 2: Recall the formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2).
---Step 3: Substitute the values of 'a' and 'b' into the formula.
---Step 4: (x - 2)(x^2 + (x)(2) + 2^2)
---Step 5: Simplify the expression inside the second bracket.
---Step 6: (x - 2)(x^2 + 2x + 4)
Answer: The factorisation of x^3 - 8 is (x - 2)(x^2 + 2x + 4).
Why It Matters
Understanding this factorisation helps simplify complex equations, which is crucial in fields like Computer Science for writing efficient code or in Engineering for designing structures. It's also used in Cryptography to secure your WhatsApp messages and online transactions, making sure your data is safe from prying eyes.
Common Mistakes
MISTAKE: Writing (a - b)(a^2 - ab + b^2) | CORRECTION: Remember the middle term in the second bracket is always +ab for difference of cubes. It's a^2 + ab + b^2.
MISTAKE: Confusing (a - b)^3 with a^3 - b^3 | CORRECTION: (a - b)^3 is (a - b)(a - b)(a - b), which expands to a^3 - 3a^2b + 3ab^2 - b^3. It is not the same as a^3 - b^3.
MISTAKE: Incorrectly finding 'a' or 'b' when numbers are involved (e.g., for 27, writing b=9) | CORRECTION: Always find the cube root. For 27, the cube root is 3 (since 3x3x3=27), not 9.
Practice Questions
Try It Yourself
QUESTION: Factorise y^3 - 1. | ANSWER: (y - 1)(y^2 + y + 1)
QUESTION: Factorise 8m^3 - 27n^3. | ANSWER: (2m - 3n)(4m^2 + 6mn + 9n^2)
QUESTION: Factorise 64x^3 - 125. | ANSWER: (4x - 5)(16x^2 + 20x + 25)
MCQ
Quick Quiz
Which of the following is the correct factorisation of p^3 - q^3?
(p - q)(p^2 - pq + q^2)
(p - q)(p^2 + pq + q^2)
(p + q)(p^2 - pq + q^2)
(p + q)(p^2 + pq + q^2)
The Correct Answer Is:
B
The correct formula for the difference of two cubes is (a - b)(a^2 + ab + b^2). Option B correctly applies this formula with 'p' as 'a' and 'q' as 'b'.
Real World Connection
In the Real World
This concept is used when engineers design bridge structures or buildings. They often deal with equations that need simplifying to find optimal dimensions or forces. For example, in computer graphics, when designing 3D models of objects like cars or buildings, understanding how to factorise and simplify complex polynomial equations helps in creating smooth curves and surfaces efficiently, just like how ISRO scientists use maths for rocket trajectories.
Key Vocabulary
Key Terms
FACTORISATION: Breaking down an expression into simpler expressions (factors) that multiply together to give the original expression. | CUBE: The result of multiplying a number by itself three times (e.g., 2^3 = 8). | IDENTITY: An equation that is true for all possible values of its variables. | ALGEBRAIC EXPRESSION: A mathematical phrase that contains numbers, variables, and operators.
What's Next
What to Learn Next
Great job learning about the factorisation of the difference of two cubes! Next, you should explore the 'Factorisation of the Sum of Two Cubes'. It's very similar but with a small change in signs, and knowing both will make you a master of cube factorisations!
