Question

if x+y+z= -1, xy+yz+zx= -1,xyz= -1 find x^3 +y^3 +z^3​

Answer 1

Answer:

⇒ x³ + y³ + z³ = - 7

Step-by-step explanation:

Given :

If,

x + y + z = -1,

xy + yz + zx = -1,

xyz = -1

Then, find : x³ + y³ + z³

Solution :

We know that,

a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ac)

By substituting,

a = x , b = y , c = z,.

⇒ x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

To equate this,

We need to find the value of : x² + y² + z²

We know that,

(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc

By substituting,

a = x , b = y , c = z,.

We get,

⇒ (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

⇒ (x + y + z)² = x² + y² + z² + 2(xy + yz + zx)

⇒ (-1)² = x² + y² + z² + 2(-1)

⇒ 1 = x² + y² + z² - 2

⇒ 1 + 2= x² + y² + z²

⇒  x² + y² + z² = 3 ...(i)

__

x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)

⇒ x³ + y³ + z³ = (x + y + z)(x² + y² + z² - (xy + yz -+ zx)) + 3xyz

⇒ x³ + y³ + z³ = (-1) ((3) - (-1)) + 3(-1)

⇒ x³ + y³ + z³ = (-1) (3 + 1) - 3

⇒ x³ + y³ + z³ = (-1) (4) - 3

⇒  x³ + y³ + z³ = - 4 - 3

⇒ x³ + y³ + z³ = - 7