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What is the Factorisation of a³+b³+c³-3abc?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The factorisation of a³+b³+c³-3abc is an important algebraic identity that helps break down this complex expression into simpler multiplied terms. It states that a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca). This identity is super useful for solving many math problems quickly.
Simple Example
Quick Example
Imagine you have three numbers, say 1, 2, and 3. If you want to calculate (1³+2³+3³-3*1*2*3), it might look tricky. Using the factorisation formula, we can first find (1+2+3) = 6. Then, we find (1²+2²+3²-1*2-2*3-3*1) = (1+4+9-2-6-3) = 14-11 = 3. So, the final answer is 6 * 3 = 18. Much easier!
Worked Example
Step-by-Step
PROBLEM: Factorise 8x³+y³+27z³-18xyz
STEP 1: Identify 'a', 'b', and 'c'. Here, 8x³ = (2x)³, so a = 2x. y³ = (y)³, so b = y. 27z³ = (3z)³, so c = 3z.
---STEP 2: Check if -3abc matches -18xyz. -3 * (2x) * (y) * (3z) = -18xyz. Yes, it matches!
---STEP 3: Apply the factorisation formula: a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca).
---STEP 4: Substitute the values of a, b, and c into the first bracket: (2x+y+3z).
---STEP 5: Substitute the values into the second bracket: ((2x)² + y² + (3z)² - (2x)(y) - (y)(3z) - (3z)(2x)).
---STEP 6: Simplify the terms in the second bracket: (4x² + y² + 9z² - 2xy - 3yz - 6zx).
---STEP 7: Combine both brackets.
ANSWER: (2x+y+3z)(4x²+y²+9z²-2xy-3yz-6zx)
Why It Matters
This identity is not just for exams; it's a foundational tool in higher mathematics and computer science. Engineers use such algebraic tools to simplify complex equations in designing bridges or coding software. Data scientists and AI/ML experts use similar principles to optimize algorithms, making apps like Google Maps or Netflix faster and smarter.
Common Mistakes
MISTAKE: Forgetting the '-3abc' term or changing its sign. | CORRECTION: The formula is specifically for a³+b³+c³-3abc. If you have a³+b³+c³+3abc, it's a different expression and cannot be factorised using this specific identity directly.
MISTAKE: Incorrectly squaring terms or multiplying 'ab', 'bc', 'ca'. Forgetting the negative signs in the second bracket. For example, writing (a²+b²+c²+ab+bc+ca). | CORRECTION: Remember, the second bracket has negative signs: (a²+b²+c²-ab-bc-ca). Always double-check your signs and calculations.
MISTAKE: Not identifying 'a', 'b', and 'c' correctly when coefficients are involved, like in 8x³. Students might take 'a' as 8x instead of 2x. | CORRECTION: Always write the cubed term as (something)³ first. For 8x³, it's (2x)³, so a=2x. For 27y³, it's (3y)³, so b=3y.
Practice Questions
Try It Yourself
QUESTION: Factorise p³+q³+r³-3pqr | ANSWER: (p+q+r)(p²+q²+r²-pq-qr-rp)
QUESTION: Factorise x³+8y³+z³-6xyz | ANSWER: (x+2y+z)(x²+4y²+z²-2xy-2yz-zx)
QUESTION: If a+b+c=0, what is the value of a³+b³+c³-3abc? | ANSWER: If a+b+c=0, then (a+b+c)(a²+b²+c²-ab-bc-ca) = 0 * (a²+b²+c²-ab-bc-ca) = 0. So, a³+b³+c³-3abc = 0. This implies a³+b³+c³ = 3abc when a+b+c=0.
MCQ
Quick Quiz
Which of these is the correct factorisation of a³+b³+c³-3abc?
(a+b+c)(a²+b²+c²+ab+bc+ca)
(a+b-c)(a²+b²+c²-ab-bc-ca)
(a+b+c)(a²+b²+c²-ab-bc-ca)
(a-b-c)(a²+b²+c²-ab-bc-ca)
The Correct Answer Is:
C
Option C correctly represents the factorisation of a³+b³+c³-3abc. The first bracket is a+b+c, and the second bracket has all squared terms positive and all product terms (ab, bc, ca) negative.
Real World Connection
In the Real World
This identity is fundamental in fields like cryptography, where complex numbers and expressions are factorised to secure online transactions, like those you make with UPI or Net Banking. Understanding how to break down large expressions into smaller parts is key to understanding how these systems work and keeping your data safe.
Key Vocabulary
Key Terms
FACTORISATION: Breaking down an expression into a product of simpler expressions | ALGEBRAIC IDENTITY: An equation that is true for all possible values of its variables | EXPRESSION: A combination of numbers, variables, and operation symbols | COEFFICIENT: A numerical or constant quantity placed before and multiplying the variable in an algebraic term
What's Next
What to Learn Next
Great job mastering this identity! Next, you should explore more advanced algebraic identities and their applications, especially those involving powers higher than three. This will further strengthen your base for complex problem-solving in mathematics and science.
