3%2B(y2-z2)3%2B(z-2-x-2)3%2F(x-y)3%2B(y-z)3%2B(z-x)3.jpeg "(x2-y2)3+(y2-z2)3+(z 2-x 2)3/(x-y)3+(y-z)3+(z-x)3")
Simplifying the Expression: (x²-y²)³ + (y²-z²)³ + (z²-x²)³ / (x-y)³ + (y-z)³ + (z-x)³
This expression appears complex, but we can simplify it using algebraic manipulation and a key factorization formula. Let's break it down step by step:
Understanding the Key Factorization
The core of simplifying this expression lies in recognizing the following factorization:
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc)
This formula holds true for any values of 'a', 'b', and 'c'.
Applying the Factorization
Let's apply this formula to our expression:
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Identify the terms: We have (x²-y²), (y²-z²), and (z²-x²) as our 'a', 'b', and 'c' respectively.
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Calculate the necessary components:
- (x²-y²) + (y²-z²) + (z²-x²) = 0
- (x²-y²)² + (y²-z²)² + (z²-x²)² = 2(x⁴ + y⁴ + z⁴ - x²y² - x²z² - y²z²)
- (x²-y²)(y²-z²)(z²-x²) = (x²-y²)(z²-x²)(y²-z²) (Note the rearrangement)
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Substitute into the factorization formula: (x²-y²)³ + (y²-z²)³ + (z²-x²)³ - 3(x²-y²)(y²-z²)(z²-x²) = (0) * [2(x⁴ + y⁴ + z⁴ - x²y² - x²z² - y²z²) - (x²-y²)(z²-x²)(y²-z²)]
Simplifying the Denominator
Now let's look at the denominator:
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Apply the difference of cubes formula:
- (x-y)³ = x³ - 3x²y + 3xy² - y³
- (y-z)³ = y³ - 3y²z + 3yz² - z³
- (z-x)³ = z³ - 3z²x + 3zx² - x³
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Add the terms together: (x-y)³ + (y-z)³ + (z-x)³ = -3(x²y - xy² + y²z - yz² + z²x - zx²)
Combining the Simplified Expressions
Our simplified expression now looks like this:
0 / -3(x²y - xy² + y²z - yz² + z²x - zx²)
Final Result
Any number divided by 0 is undefined. Therefore, the simplified form of the given expression is undefined.
It's important to note that this expression is undefined for any values of x, y, and z where the denominator (x²y - xy² + y²z - yz² + z²x - zx²) equals 0.