Question

$x+y+z =1 ,xy+yz+zx=1 and xyz=-1 then the find the value of x^{3} +y^{3}+ z^{3]$

Answer 1

Answer:

The value of x^3 + y^3 + z^3 = 1

Step-by-step explanation:

Given that:-

• x + y + z = 1

• xy + yz +zx = 1

• xyz = 1

To find:-

The value of x^3 + y^3 + z^3 = ?

Solution:-

We know that x^3 + y^3 + z^3 – 3 xyz = (x + y + z) (x^2 + y^2 + z^2 – xy – yz – zx)

=> x^3 + y^3 + z^3 – 3xyz

=> (x + y + z) (x^2 + y^2 + z^2 + 2xy + 2yz + 2zx – 3xy – 3yz – 3zx)

(Subtracting and adding 2xy + 2yz + 2zx)

=> x^3 + y^3 + z^3 – 3xyz = (x + y + z) {(x + y + z)2 – 3(xy + yz + zx)}

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

[Putting x + y + z = 1; xy + yz + zx = -1; xyz = -1]

=> x^3 + y^3 + z^3 + 3 = 4

=> x^3 + y^3 + z^3 = 4 – 3

=> x^3 + y^3 + z^3 = 1

Answer:-

Tha value of x^3 + y^3 + z^3 = 1

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