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If x, y, and z are said to be the real numbers, then what is the value of (x - y)3 + (y - z)3 + (z - x)3 ?
- a)0
- b)(x - y)
- c)3(x - y) (y - z) (z - x)
- d)(x - y) (y - z) (z - x)
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If x, y, and z are said to be the real numbers, then what is the value...
Suppose a = (x - y), b = (y - z), and c = (z - x)
On adding a, b, and c we will get
a + b + c = x -y + y - z + z - x
=> a + b + c =0
So, a3 + b3 + c3 = 3abc [because if a + b + c = 0, then a3 + b3 + c3 = 3abc]
We can say that (x - y)3 + (y - z)3 + (z - x)3 = 3 (x - y) (y - z) (z - x)
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If x, y, and z are said to be the real numbers, then what is the value...
Explanation:
Given Expression: (x - y)^3 + (y - z)^3 + (z - x)^3
Expand the Cubes:
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
(y - z)^3 = y^3 - 3y^2z + 3yz^2 - z^3
(z - x)^3 = z^3 - 3z^2x + 3zx^2 - x^3
Substitute back into the Given Expression:
(x - y)^3 + (y - z)^3 + (z - x)^3
= (x^3 - 3x^2y + 3xy^2 - y^3) + (y^3 - 3y^2z + 3yz^2 - z^3) + (z^3 - 3z^2x + 3zx^2 - x^3)
= x^3 - 3x^2y + 3xy^2 - y^3 + y^3 - 3y^2z + 3yz^2 - z^3 + z^3 - 3z^2x + 3zx^2 - x^3
= x^3 - y^3 + y^3 - z^3 + z^3 - x^3
= 0
Therefore, the value of the expression (x - y)^3 + (y - z)^3 + (z - x)^3 is 0. Hence, the correct answer is option 'a) 0'.
Given Expression: (x - y)^3 + (y - z)^3 + (z - x)^3
Expand the Cubes:
(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3
(y - z)^3 = y^3 - 3y^2z + 3yz^2 - z^3
(z - x)^3 = z^3 - 3z^2x + 3zx^2 - x^3
Substitute back into the Given Expression:
(x - y)^3 + (y - z)^3 + (z - x)^3
= (x^3 - 3x^2y + 3xy^2 - y^3) + (y^3 - 3y^2z + 3yz^2 - z^3) + (z^3 - 3z^2x + 3zx^2 - x^3)
= x^3 - 3x^2y + 3xy^2 - y^3 + y^3 - 3y^2z + 3yz^2 - z^3 + z^3 - 3z^2x + 3zx^2 - x^3
= x^3 - y^3 + y^3 - z^3 + z^3 - x^3
= 0
Therefore, the value of the expression (x - y)^3 + (y - z)^3 + (z - x)^3 is 0. Hence, the correct answer is option 'a) 0'.
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