What is the Expansion of (x+y+z)³?

S3-SA1-0732

Grade Level:

Class 8

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

Definition

What is it?

The expansion of (x+y+z)³ is the result of multiplying the expression (x+y+z) by itself three times. It gives a long algebraic expression with many terms that combine powers of x, y, and z. This formula helps simplify complex calculations involving three variables.

Simple Example

Quick Example

Imagine you have three friends, Rohan, Priya, and Sameer, and each of them has some amount of money. If you want to calculate the total amount of money they have, and then cube that total, it's like finding (Rohan's money + Priya's money + Sameer's money)³. The expansion shows all the different ways their individual amounts combine to form that cube.

Worked Example

Step-by-Step

Let's expand (a+b+c)³.

First, we know (A+B)² = A² + B² + 2AB. We can think of (a+b+c) as ( (a+b) + c ).

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Step 1: Expand ( (a+b) + c )² = (a+b)² + c² + 2(a+b)c
= a² + b² + 2ab + c² + 2ac + 2bc

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Step 2: Now, (a+b+c)³ = (a+b+c) * (a+b+c)²
= (a+b+c) * (a² + b² + c² + 2ab + 2bc + 2ca)

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Step 3: Multiply each term from (a+b+c) with the second bracket.
Multiply by 'a': a(a² + b² + c² + 2ab + 2bc + 2ca) = a³ + ab² + ac² + 2a²b + 2abc + 2a²c

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Step 4: Multiply by 'b': b(a² + b² + c² + 2ab + 2bc + 2ca) = a²b + b³ + bc² + 2ab² + 2b²c + 2abc

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Step 5: Multiply by 'c': c(a² + b² + c² + 2ab + 2bc + 2ca) = a²c + b²c + c³ + 2abc + 2bc² + 2c²a

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Step 6: Combine all the terms from Steps 3, 4, and 5:
a³ + ab² + ac² + 2a²b + 2abc + 2a²c
+ a²b + b³ + bc² + 2ab² + 2b²c + 2abc
+ a²c + b²c + c³ + 2abc + 2bc² + 2c²a

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Step 7: Group like terms:
a³ + b³ + c³
+ (ab² + 2ab²) + (ac² + 2ac²) + (a²b + 2a²b)
+ (bc² + 2bc²) + (b²c + 2b²c) + (2abc + 2abc + 2abc)

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Step 8: Simplify:
a³ + b³ + c³ + 3a²b + 3ab² + 3b²c + 3bc² + 3c²a + 3ca² + 6abc

Answer: (a+b+c)³ = a³ + b³ + c³ + 3a²b + 3ab² + 3b²c + 3bc² + 3c²a + 3ca² + 6abc

Why It Matters

Understanding expansions like (x+y+z)³ is crucial in fields like Engineering and Physics for modeling complex systems. Data Scientists use similar polynomial expressions to fit data and make predictions. In Cryptography, these foundational algebraic principles help create secure codes for online transactions, keeping your UPI payments safe!

Common Mistakes

MISTAKE: Students often forget the 'c' term when applying (A+B)² to ( (a+b)+c )², thinking it's just (a+b)² + c². | CORRECTION: Remember the formula for (A+B)² is A² + B² + 2AB. So, when A=(a+b) and B=c, it becomes (a+b)² + c² + 2(a+b)c.

MISTAKE: Incorrectly combining like terms or missing some terms when multiplying out the full expansion. | CORRECTION: Be systematic. Multiply each term from the first bracket by every term in the second bracket. Then, carefully list all terms and group identical terms together before adding their coefficients.

MISTAKE: Confusing (x+y+z)³ with x³+y³+z³. | CORRECTION: Remember that cubing a sum is not the same as summing the cubes. (x+y+z)³ means (x+y+z) multiplied by itself three times, which generates many cross-product terms, not just the individual cubed terms.

Practice Questions

Try It Yourself

QUESTION: Expand (p+q+r)³ | ANSWER: p³ + q³ + r³ + 3p²q + 3pq² + 3q²r + 3qr² + 3r²p + 3rp² + 6pqr

QUESTION: Find the expansion of (2a+b+c)³ | ANSWER: 8a³ + b³ + c³ + 12a²b + 6ab² + 3b²c + 3bc² + 12a²c + 6ac² + 12abc

QUESTION: If (x+y+z) = 5 and (xy+yz+zx) = 7, find the value of (x³+y³+z³ - 3xyz) + 3xy(x+y) + 3yz(y+z) + 3zx(z+x). Hint: Relate to the expansion of (x+y+z)³. | ANSWER: 125

MCQ

Quick Quiz

Which of the following is NOT a term in the expansion of (x+y+z)³?

x³

3x²y

6xyz

3xyz²

The Correct Answer Is:

The expansion of (x+y+z)³ includes terms like x³, y³, z³, 3x²y, 3xy², 3y²z, 3yz², 3z²x, 3zx², and 6xyz. The term 3xyz² is not part of this standard expansion.

Real World Connection

In the Real World

In Computer Science, especially in graphics and game development, equations involving three variables (like x, y, z for 3D coordinates) are common. When designing a new building or a complex bridge using CAD software, engineers use these types of expansions to model volumes and stress, ensuring the structure is stable and safe, just like ISRO scientists use complex math for rocket trajectories.

Key Vocabulary

Key Terms

EXPANSION: Writing out an algebraic expression as a sum of its individual terms after multiplication | TERM: A single number, variable, or product of numbers and variables | COEFFICIENT: The numerical factor of a term | ALGEBRAIC IDENTITY: An equation that is true for all possible values of its variables

What's Next

What to Learn Next

Now that you've mastered (x+y+z)³, you're ready to explore more complex polynomial expansions and factorizations. Understanding these building blocks will make advanced algebra and calculus much easier, opening doors to exciting careers in technology and science!