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Algebraic Identities

If x2+y2+z2= ...

Question

If $(x^{2} + y^{2} + z^{2}) = (x y + y z + z x)$ , $what is the value of x^{3} + y^{3} + z^{3}$ ?

A

$3 x y z$

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B

$- 3 x y z$

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C

$2 x y z$

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D

$- 2 x y z$

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Solution

The correct option is A
$3 x y z$

We know that

$x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x) - - - (1)$

According to question, $(x^{2} + y^{2} + z^{2}) = (x y + y z + z x)$ .
$∴$ RHS will be zero in equation (1)

So, $x^{3} + y^{3} + z^{3} = 3 x y z$

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Similar questions

If $(x^{2} + y^{2} + z^{2}) = (x y + y z + z x)$ , $what is the value of x^{3} + y^{3} + z^{3}$ ?

\frac{x^{3} + y^{3} + z^{3} - 3 x y z}{a^{3} + b^{3} + c^{3} - 3 a b c} \times \frac{a^{2} + b^{2} + c^{2} - a b - b c - c a}{x^{2} + y^{2} + z^{2} - x y - y z - z x}

x^{2} + y^{2} + z^{2} = x y + y z + z x

and the product of

x, y

z

27

, find the value of

x^{3} + y^{3} + z^{3}

Q. Using the identity and proof:

x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)

.

x^{3} + y^{3} + z^{3} - 3 x y z \geq (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)

.
State whether the above statement is true or false.

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