Question

if x + y + z =1,xy+yz+zx=2 and xyz=-2 find the value of x³+y³+z³ . please answer​

Answer 1

Step-by-step explanation:

We know that,

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

Or

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

We have everything except x² + y² + z² to find x³ + y³ + z³

So first we will find the value of x² + y² + z² In order to find x³ + y³ + z³

Given:

• x + y + z = 1
• xy + yz + xz = 2
• xyz = -2

We know that,

→ (x + y + z)² = x² + y² + z² + 2(xy + yz + xz)

Substituting the values, we get

→ (1)² = x² + y² + z² + 2(2)

→ 1 = x² + y² + z² + 4

→ x² + y² + z² = 1 - 4

→ x² + y² + z² = -3

Now,

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

Substituting the values, we get

→ x³ + y³ + z³ - 3(-2) = (1)[-3- (2)]

→ x³ + y³ + z³ + 6 = -3 - 2

→ x³ + y³ + z³ + 6 = -5

→ x³ + y³ + z³ = -5 - 6

→ x³ + y³ + z³ = -11

Hence, x³ + y³ + z³ = -11