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What is the Factorisation of Sum of Two Cubes?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
Factorisation of the sum of two cubes is a special algebraic identity used to break down an expression where two terms, each raised to the power of 3, are added together. It helps simplify complex expressions into a product of simpler factors. The general form is a^3 + b^3 = (a + b)(a^2 - ab + b^2).
Simple Example
Quick Example
Imagine you have two cube-shaped boxes. One has a side of 3 cm (so its volume is 3^3 = 27 cubic cm) and another has a side of 2 cm (so its volume is 2^3 = 8 cubic cm). If you add their volumes, you get 27 + 8 = 35. Using factorisation, we can write this sum in a different, structured way, even for algebraic terms like x^3 + y^3.
Worked Example
Step-by-Step
Let's factorise the expression x^3 + 8.
Step 1: Identify the two terms being cubed. Here, the first term is x^3. The second term is 8, which can be written as 2^3.
---Step 2: So, we have x^3 + 2^3. Comparing this to the identity a^3 + b^3, we can see that a = x and b = 2.
---Step 3: Recall the factorisation formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).
---Step 4: Substitute 'a' with 'x' and 'b' with '2' into the formula.
---Step 5: The first part (a + b) becomes (x + 2).
---Step 6: The second part (a^2 - ab + b^2) becomes (x^2 - (x)(2) + 2^2).
---Step 7: Simplify the second part: (x^2 - 2x + 4).
---Step 8: Combine the parts. So, x^3 + 8 = (x + 2)(x^2 - 2x + 4).
Answer: (x + 2)(x^2 - 2x + 4)
Why It Matters
Understanding this factorisation helps engineers design structures, computer scientists write efficient code for graphics, and even helps in data analysis to simplify complex equations. It's a foundational skill for solving problems in fields like AI/ML and Physics, opening doors to careers in technology and research.
Common Mistakes
MISTAKE: Writing the middle term of the second factor as +ab instead of -ab (e.g., (a+b)(a^2 + ab + b^2)). | CORRECTION: Remember, for the sum of cubes (a^3 + b^3), the middle term in the trinomial factor is always negative (-ab).
MISTAKE: Forgetting to square the 'b' term in the second factor (e.g., (a+b)(a^2 - ab + b)). | CORRECTION: Always remember to square both 'a' and 'b' in the second factor, so it should be a^2 and b^2.
MISTAKE: Incorrectly identifying 'a' or 'b' when a number is involved (e.g., for x^3 + 27, thinking b=27 instead of b=3). | CORRECTION: Always find the cube root of the numerical term to correctly identify 'b'. For 27, the cube root is 3, so b=3.
Practice Questions
Try It Yourself
QUESTION: Factorise y^3 + 64. | ANSWER: (y + 4)(y^2 - 4y + 16)
QUESTION: Factorise 8m^3 + 125n^3. | ANSWER: (2m + 5n)(4m^2 - 10mn + 25n^2)
QUESTION: Factorise (x+1)^3 + 8. | ANSWER: (x + 3)(x^2 + x + 7)
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:
C
The correct identity for the sum of two cubes (a^3 + b^3) is (a + b)(a^2 - ab + b^2). Option C matches this form with p and q.
Real World Connection
In the Real World
In computer graphics, when designing 3D models of objects like cars or buildings, complex shapes are often broken down into simpler geometric forms. Factorisation of cubes can be used in algorithms to simplify the mathematical equations describing these shapes, making rendering faster and more efficient for games or architectural visualisations.
Key Vocabulary
Key Terms
FACTORISATION: Breaking down an expression into a product of simpler expressions | CUBE: A number or term multiplied by itself three times (e.g., x^3) | IDENTITY: An equation that is true for all possible values of its variables | BINOMIAL: An algebraic expression with two terms | TRINOMIAL: An algebraic expression with three terms
What's Next
What to Learn Next
Great job learning about the sum of two cubes! Next, you should explore the 'Factorisation of Difference of Two Cubes' (a^3 - b^3). It's very similar, but with a few key sign changes, and understanding both will make you a master of cubic factorisation!
