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Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)?
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Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz ...
Proof:
To prove the given statement, we need to show that:
x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
Expanding the left side:
x^3 * y^3 * z^3 - 3xyz
= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * 3
= (x * y * z) * (x^2 * y^2 * z^2 - 3)
Expanding the right side:
(x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * (xy + xz + yz)
= (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)
Comparing both sides:
We can see that the expanded forms of both sides are the same:
(x * y * z) * (x^2 * y^2 * z^2 - 3) = (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)
Since both sides of the equation are equal, we can conclude that:
x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
Therefore, the given statement is proved.
Summary:
By expanding and comparing the left and right sides of the equation, we have shown that x^3 * y^3 * z^3 - 3xyz is equal to (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz). Hence, the given statement is proved.
To prove the given statement, we need to show that:
x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
Expanding the left side:
x^3 * y^3 * z^3 - 3xyz
= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * 3
= (x * y * z) * (x^2 * y^2 * z^2 - 3)
Expanding the right side:
(x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
= (x * y * z) * (x^2 * y^2 * z^2) - (x * y * z) * (xy + xz + yz)
= (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)
Comparing both sides:
We can see that the expanded forms of both sides are the same:
(x * y * z) * (x^2 * y^2 * z^2 - 3) = (x * y * z) * (x^2 * y^2 * z^2) - (x^2 * y * z^2 + x * y^2 * z^2 + x * y * z^2)
Since both sides of the equation are equal, we can conclude that:
x^3 * y^3 * z^3 - 3xyz = (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz)
Therefore, the given statement is proved.
Summary:
By expanding and comparing the left and right sides of the equation, we have shown that x^3 * y^3 * z^3 - 3xyz is equal to (x * y * z) * (x^2 * y^2 * z^2 - xy - xz - yz). Hence, the given statement is proved.
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Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)?
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Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)?.
Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)? for Class 9 2024 is part of Class 9 preparation. The Question and answers have been prepared according to the Class 9 exam syllabus. Information about Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)? covers all topics & solutions for Class 9 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Prove: x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz)?.
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