Question

If x+y+z=0 then x^3+y^3+z^3 is
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Answer 1

Answer:

Step-by-step explanation:

if x+y+z = 0

then x³+y³+z³ = 3xyz

Answer 2

Question :

If $\bf{x + y + z = 0}$ , then find the value of $\bf{x^{3} + y^{3} + z^{3}}$

Given :

$\underline{\bf{x + y + z = 0}}$

To Find :

$\underline{\boxed{\bf{x + y + z = 0}}}$

Solution :

Given Equation :

By subtracting (z) from both the sides of the Equation , we get :

$\implies \bf{x + y + z - z = 0 - z}$

$\implies \bf{x + y + \not{z} - \not{z} = 0 - z}$

$\implies \bf{x + y = 0 - z}$

$\implies \bf{x + y = - z}$

Now , by cubing on both the sides , we get :

$\implies \bf{(x + y)^{3} = (- z)^{3}}$

But we know that :

$\bf{(a + b)^{3} = a^{3} + b^{3} + 3ab(a + b)}$

So , by substituting this value of $\bf{(a + b)^{3}}$ , we get :

$\implies \bf{x^{3} + x^{3} + 3xy(x + y) = (- z)^{3}}$

From the Equation $\bf{x + y + z = 0}$ , we get :

⠀⠀⠀$:\implies \bf{x + y + z = 0}$

⠀⠀⠀$:\implies \bf{x + y = - z}$

Hence, the value of (x + y) is -z .

Now , putting the value of (x + y) in the equation , we get :

$\implies \bf{x^{3} + x^{3} + 3xy(-z) = (- z)^{3}}$

$\implies \bf{x^{3} + x^{3} - 3xyz = (- z)^{3}}$

Now , by adding (z³) on both the sides , we get :

$\implies \bf{x^{3} + x^{3} - 3xyz + z^{3} = (- z)^{3} + z^{3}}$

$\implies \bf{x^{3} + x^{3} - 3xyz + z^{3} = (- \not{z})^{3} + \not{z}^{3}}$

$\implies \bf{x^{3} + x^{3} - 3xyz + z^{3} = 0}$

Then , by adding (3xyz) in both the sides , we get :

$\implies \bf{x^{3} + x^{3} - 3xyz + z^{3} + 3xyz = 0 + 3xyz}$

$\implies \bf{x^{3} + x^{3} + z^{3}= 3xyz}$

$\underline{\therefore \bf{x^{3} + x^{3} + z^{3}= 3xyz}}$

Hence, the value of (x³ + y³ + z³) is 3xyz.