What is (a+b+c)³? | Simple Explanation for Class 6

S3-SA1-0553

What is the Identity (a+b+c)³?

Grade Level:

Class 6

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

Definition

What is it?

The identity (a+b+c)³ is a special formula in algebra that helps us quickly expand or multiply a sum of three terms (a, b, and c) by itself three times. It's like a shortcut to find the value of (a+b+c) * (a+b+c) * (a+b+c) without doing long multiplication.

Simple Example

Quick Example

Imagine you have three friends, A, B, and C, and each has a certain number of marbles. If you want to find the 'cube' of their total marbles, (A+B+C)³, this identity gives you a ready-made formula. For example, if A has 1 marble, B has 2, and C has 3, then (1+2+3)³ = 6³ = 216.

Worked Example

Step-by-Step

Let's expand (x+2y+3z)³ using the identity (a+b+c)³ = a³ + b³ + c³ + 3(a+b)(b+c)(c+a) + 6abc.
---Step 1: Identify a, b, and c. Here, a = x, b = 2y, c = 3z.
---Step 2: Substitute these values into the identity.
(x+2y+3z)³ = x³ + (2y)³ + (3z)³ + 3(x+2y)(2y+3z)(3z+x) + 6(x)(2y)(3z)
---Step 3: Simplify the cubed terms.
x³ + 8y³ + 27z³
---Step 4: Simplify the last term (6abc).
6(x)(2y)(3z) = 36xyz
---Step 5: Combine the simplified terms and the remaining product term. The full expansion is:
x³ + 8y³ + 27z³ + 3(x+2y)(2y+3z)(3z+x) + 36xyz
---Answer: The expanded form of (x+2y+3z)³ is x³ + 8y³ + 27z³ + 3(x+2y)(2y+3z)(3z+x) + 36xyz.

Why It Matters

Understanding algebraic identities like (a+b+c)³ helps engineers design complex structures and computer scientists optimize algorithms. Data scientists use similar principles to model relationships in large datasets, making predictions from cricket scores to market trends. It's a foundational skill for many exciting careers!

Common Mistakes

MISTAKE: Students often forget the 'cross-product' terms and only cube individual terms, writing (a+b+c)³ = a³ + b³ + c³. | CORRECTION: Remember that cubing a sum is not just cubing each part; there are many more terms involved, like 3(a+b)(b+c)(c+a) and 6abc.

MISTAKE: Incorrectly expanding terms like (2y)³ as 2y³ instead of 8y³. | CORRECTION: When a term like '2y' is cubed, both the number and the variable are cubed: (2y)³ = 2³ * y³ = 8y³.

MISTAKE: Misremembering the complex middle terms of the identity. | CORRECTION: While the full identity is long, focusing on the common factors and pattern (like 3(a+b)(b+c)(c+a) + 6abc) helps in recall. Practice helps a lot!

Practice Questions

Try It Yourself

QUESTION: Expand (p+q+r)³. | ANSWER: p³ + q³ + r³ + 3(p+q)(q+r)(r+p) + 6pqr

QUESTION: If a=1, b=2, c=0, what is (a+b+c)³? | ANSWER: (1+2+0)³ = 3³ = 27

QUESTION: Expand (m+2n+1)³. | ANSWER: m³ + 8n³ + 1 + 3(m+2n)(2n+1)(1+m) + 6m(2n)(1) = m³ + 8n³ + 1 + 3(m+2n)(2n+1)(1+m) + 12mn

MCQ

Quick Quiz

Which of the following is the correct expansion for (x+y+z)³?

x³ + y³ + z³

x³ + y³ + z³ + 3xyz

x³ + y³ + z³ + 3(x+y)(y+z)(z+x) + 6xyz

x³ + y³ + z³ + 3(x+y+z)

The Correct Answer Is:

Option C is the complete and correct identity for (x+y+z)³. Options A, B, and D miss many important terms that arise from the multiplication of three binomials.

Real World Connection

In the Real World

In designing a new multi-storey building in India, civil engineers use complex algebraic equations that often involve cubing sums of different dimensions or forces. Similarly, in computer graphics, calculating the volume of 3D objects made of multiple parts can involve similar expansions to ensure accurate rendering and simulations.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | EXPANSION: The process of multiplying out the terms of an expression. | CUBED: A number or expression multiplied by itself three times. | ALGEBRAIC EXPRESSION: A combination of variables, numbers, and arithmetic operations.

What's Next

What to Learn Next

Now that you understand (a+b+c)³, you can explore other cubic identities like (a+b)³ and (a-b)³. These identities are building blocks for solving more complex polynomial equations and understanding higher-level algebra.