What is the Identity (x+y+z)(x²+y²+z²-xy-yz-zx)? | Simple Explanation for Class 7

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What is the Identity (x+y+z)(x²+y²+z²-xy-yz-zx) = x³+y³+z³-3xyz?

Grade Level:

Class 7

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

Definition

What is it?

The identity (x+y+z)(x^2+y^2+z^2-xy-yz-zx) = x^3+y^3+z^3-3xyz is a special algebraic formula that helps simplify complex expressions. It shows a unique relationship between the sum of three numbers and the sum of their cubes, minus three times their product.

Simple Example

Quick Example

Imagine you have three friends, A, B, and C, and you want to calculate a special 'friendship score'. If A=1, B=2, C=3, and you use this formula, it helps you quickly find a value without lots of small multiplications. It's like a shortcut to a big calculation!

Worked Example

Step-by-Step

Let's verify the identity with x=1, y=2, and z=3.

Step 1: Calculate the Left Hand Side (LHS).
LHS = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)
LHS = (1+2+3)(1^2+2^2+3^2 - (1*2) - (2*3) - (3*1))
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Step 2: Simplify the first bracket.
(1+2+3) = 6
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Step 3: Simplify the second bracket.
(1+4+9 - 2 - 6 - 3) = (14 - 11) = 3
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Step 4: Multiply the results.
LHS = 6 * 3 = 18
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Step 5: Calculate the Right Hand Side (RHS).
RHS = x^3+y^3+z^3-3xyz
RHS = 1^3+2^3+3^3 - 3*(1*2*3)
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Step 6: Simplify the RHS.
RHS = 1+8+27 - 3*6
RHS = 36 - 18 = 18
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Step 7: Compare LHS and RHS.
Since LHS = 18 and RHS = 18, the identity is verified.
Answer: The identity holds true for x=1, y=2, z=3.

Why It Matters

This identity is super useful in fields like Computer Science and Engineering to optimize calculations, making programs run faster. Data Scientists use similar identities to simplify complex data models. Even in Physics, understanding these relationships can help solve problems related to forces and energy, opening doors to careers in technology and research.

Common Mistakes

MISTAKE: Forgetting the '-3xyz' term on the RHS or getting the signs wrong in the second bracket on the LHS. | CORRECTION: Always remember the full identity. The second bracket on the LHS has all products (xy, yz, zx) with negative signs.

MISTAKE: Confusing (x+y+z)^3 with x^3+y^3+z^3-3xyz. | CORRECTION: (x+y+z)^3 is a different expansion. This identity relates the sum of three numbers to the sum of their cubes, not the cube of their sum.

MISTAKE: Incorrectly squaring numbers or multiplying terms, especially with negative signs. | CORRECTION: Pay close attention to basic arithmetic. Double-check squares (e.g., 3^2=9, not 6) and products (e.g., 3*2=6).

Practice Questions

Try It Yourself

QUESTION: If x=2, y=3, z=1, find the value of x^3+y^3+z^3-3xyz using the identity. | ANSWER: 36

QUESTION: If x+y+z = 5 and x^2+y^2+z^2-xy-yz-zx = 7, what is the value of x^3+y^3+z^3-3xyz? | ANSWER: 35

QUESTION: Given that x+y+z=0, what can you say about the relationship between x^3+y^3+z^3 and 3xyz? | ANSWER: If x+y+z=0, then x^3+y^3+z^3-3xyz = 0, which means x^3+y^3+z^3 = 3xyz.

MCQ

Quick Quiz

Which of the following is equivalent to (a+b+c)(a^2+b^2+c^2-ab-bc-ca)?

a^3+b^3+c^3+3abc

a^3+b^3+c^3-3abc

(a+b+c)^3

a^2+b^2+c^2

The Correct Answer Is:

The correct identity is (a+b+c)(a^2+b^2+c^2-ab-bc-ca) = a^3+b^3+c^3-3abc. Options A, C, and D represent different algebraic expressions.

Real World Connection

In the Real World

This identity helps engineers design more efficient algorithms for tasks like 3D graphics rendering in video games or optimizing resource allocation in cloud computing. Imagine an app like Google Maps calculating the shortest route; simplified calculations are key to making it fast. Even financial analysts use similar algebraic principles to model complex economic scenarios.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | ALGEBRAIC EXPRESSION: A mathematical phrase that can contain numbers, variables, and operation symbols. | CUBE: The result of multiplying a number by itself three times (e.g., x*x*x or x^3). | VARIABLE: A symbol, usually a letter, representing an unknown value. | SIMPLIFY: To reduce an expression to its simplest form.

What's Next

What to Learn Next

Next, you can explore other important algebraic identities like (a+b)^3 and (a-b)^3. Understanding these will give you even more tools to solve complex problems and build a strong foundation for higher-level mathematics.