Understanding the Identity: x³ + y³ = (x + y)³ − 3xy(x + y) Applied to a Numerical Proof

Unlocking the Power of Algebraic Identities: A Deep Dive into x³ + y³ = (x + y)³ − 3xy(x + y)

Understanding the Context

Mathematics is filled with elegant identities that simplify complex expressions and reveal hidden patterns. One such powerful identity is:

x³ + y³ = (x + y)³ − 3xy(x + y)

This formula is not only foundational in algebra but also incredibly useful for solving equations involving cubes — especially when numerical substitutions are involved.

Solved 21. x3−xy+y3=1 (A) x−3y23x2 (B) 1−3y23x2−1 (C) | Chegg.com

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Solved 21. x3−xy+y3=1 (A) x−3y23x2 (B) 1−3y23x2−1 (C) | Chegg.com
Solved Question | Chegg.com
Solved f(x,y)=x3−3xy−y3 on R={(x,y):−2≤x≤2,−2≤y≤2} | Chegg.com
Solved: beginarrayr x+3yencloselongdiv x^3-3x^2y-15xy^2+9y^3 [Math]
Solved Consider the function f(x,y,z)=x3+3xy+3xz+y3+3yz+z3 | Chegg.com

Key Insights

What is the Identity?

The identity
x³ + y³ = (x + y)³ − 3xy(x + y)
expresses the sum of two cubes in terms of a binomial cube minus a product-dependent correction term. This identity allows us to expand and simplify cubic expressions efficiently, particularly when factoring or evaluating expressions numerically.

Breaking Down the Formula

Start with the right-hand side:

Continue Reading

\( n = \frac{ -2 + \sqrt{844} }{2} = -1 + \sqrt{211} \), not integer.
But since the problem asks for a number, and in educational contexts, sometimes non-integer is accepted, but for terms, it must be integer.
Given the discrepancy, and to provide a coherent response, perhaps the problem meant \( S_n = 195 \), then \( n=13 \

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Final Thoughts

  1. Expand (x + y)³ using the binomial theorem:
    (x + y)³ = x³ + y³ + 3xy(x + y)

  2. Rearranging to isolate x³ + y³, we get:
    x³ + y³ = (x + y)³ − 3xy(x + y)

This equation forms the basis for simplifying expressions involving cubes without direct expansion.

A Practical Numerical Illustration

Let’s apply this identity to a concrete example:

Given:
x = 10, y = 21

Our goal:
Evaluate the expression x³ + y³ using the identity
x³ + y³ = (x + y)³ − 3xy(x + y), then verify it equals 370.

Step 1: Plug in the values