Math Problem Statement
[(x-y)^3 + (y-z)^3 + (z-x)^3]/9((x-y)(y-z)(z-x) = ?
Solution
We are given the expression:
We need to simplify it. A known identity for this type of expression is:
This is a symmetric identity, and it greatly simplifies the numerator.
Substituting this into the original expression, we get:
The terms in the numerator and denominator cancel out, leaving:
Thus, the simplified expression is:
Would you like further details or explanations? Here are some related questions you might find interesting:
- How can we derive the identity ?
- What are some applications of symmetric polynomials in algebra?
- How does factorization help in simplifying complex expressions like this one?
- Can this method be applied to more complex cubic expressions?
- What other known identities simplify polynomials of degree three?
Tip: When simplifying expressions, always check if any known identities apply—they can save a lot of time!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Symmetric Polynomials
Identity Simplification
Formulas
(x - y)^3 + (y - z)^3 + (z - x)^3 = 3(x - y)(y - z)(z - x)
Theorems
Symmetry Identity for Cubic Expressions
Suitable Grade Level
Grades 10-12
Related Recommendation
Simplify the expression (x^3 + y^3 + z^3 - 3xyz) over ((x - y)^2 + (y - z)^2 + (z - x)^2)
Prove the Identity (x^2 - yz)^3 + (y^2 - 2x)^3 + (z^2 - xy)^3 - 3(x^2 - yz)(y^2 - 2x)(z^2 - xy) = x^3 + y^3 + z^3 - 3xyz
Prove and Solve Symmetric Cubic Expression as a Perfect Square
Simplify the Expression (x - y)(x² + xy + y²) + (y - z)(y² + yz + z²) + (z - x)(z² + xz + x²)
Simplifying x³ + y³ / (x - y) divided by Polynomial Expressions