Inaugurated by IN-SPACe
ISRO Registered Space Tutor
S3-SA1-0298
What is the Identity a³+b³+c³-3abc?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The identity a³+b³+c³-3abc is a special algebraic formula that helps us simplify expressions involving cubes of three different numbers. It states that a³+b³+c³-3abc is equal to (a+b+c)(a²+b²+c²-ab-bc-ca). This identity is especially useful when the sum (a+b+c) is zero, as it simplifies further.
Simple Example
Quick Example
Imagine you have three friends, A, B, and C. If their ages are a, b, and c, and you want to calculate a³+b³+c³-3abc, this identity gives you a shortcut. For instance, if A is 2 years old, B is 3 years old, and C is -5 years old (negative age is just for math, not real life!), and we find that a+b+c = 2+3+(-5) = 0, then we immediately know that a³+b³+c³ = 3abc, making the calculation much simpler.
Worked Example
Step-by-Step
Let's find the value of x³+y³+z³-3xyz when x=1, y=2, and z=3.
1. First, identify a, b, and c. Here, a=x=1, b=y=2, c=z=3.
2. Calculate (a+b+c): 1+2+3 = 6.
3. Calculate (a²+b²+c²-ab-bc-ca):
a² = 1² = 1
b² = 2² = 4
c² = 3² = 9
ab = 1*2 = 2
bc = 2*3 = 6
ca = 3*1 = 3
So, (1+4+9-2-6-3) = (14-11) = 3.
4. Multiply the two parts: (a+b+c) * (a²+b²+c²-ab-bc-ca) = 6 * 3 = 18.
5. Therefore, x³+y³+z³-3xyz = 18.
Answer: 18
Why It Matters
This identity is a building block for solving complex problems in fields like Computer Science and Engineering, where calculations involving multiple variables are common. It helps programmers optimize code and engineers design efficient systems. Understanding such identities can even help you think like a data scientist, making calculations faster and more accurate.
Common Mistakes
MISTAKE: Forgetting the '-3abc' part and just thinking a³+b³+c³ is the identity. | CORRECTION: The full identity is a³+b³+c³-3abc. Always remember the '-3abc' term.
MISTAKE: Mixing up the signs in the second bracket, especially writing '+ab+bc+ca' instead of '-ab-bc-ca'. | CORRECTION: The second bracket is (a²+b²+c²-ab-bc-ca). All the product terms (ab, bc, ca) have a minus sign.
MISTAKE: Assuming that if a+b+c=0, then a³+b³+c³-3abc = 0, but forgetting the special case result. | CORRECTION: If a+b+c=0, then a³+b³+c³-3abc = 0. This also means a³+b³+c³ = 3abc. Remember this powerful shortcut!
Practice Questions
Try It Yourself
QUESTION: If a=1, b=0, c=2, what is a³+b³+c³-3abc? | ANSWER: 9
QUESTION: Find the value of 5³+2³+(-7)³-3(5)(2)(-7). | ANSWER: 0 (Hint: Check if a+b+c=0 first!)
QUESTION: If x+y+z=10 and x²+y²+z²-xy-yz-zx=5, find the value of x³+y³+z³-3xyz. | ANSWER: 50
MCQ
Quick Quiz
Which of the following is equivalent to 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:
B
Option B is the correct expansion of the identity a³+b³+c³-3abc. The first bracket is the sum of the terms, and the second bracket contains the squares of the terms minus the products of pairs.
Real World Connection
In the Real World
In computer graphics and animation, engineers use complex algebraic identities like this to calculate 3D shapes and movements efficiently. For example, when designing a new character for a video game or an animated film, calculations involving many points (coordinates) in 3D space often get simplified using such formulas to make the animations smooth and realistic, just like how ISRO scientists use advanced math for rocket trajectories.
Key Vocabulary
Key Terms
IDENTITY: An equation that is true for all possible values of the variables | CUBE: A number multiplied by itself three times (e.g., 2³ = 2*2*2) | VARIABLE: A letter (like a, b, c) used to represent an unknown number | EXPRESSION: A combination of numbers, variables, and operation signs
What's Next
What to Learn Next
Now that you understand this identity, you can explore other algebraic identities like (a+b+c)² or (x+y)³. These build upon the basic concepts you've learned and will help you tackle even more challenging algebraic problems in higher classes.
