Question

if x + y + z = 14 and xy + yz + zx = 7 then find x³ + y³ + z³ - 3xyz ​

Answer 1

Given:

• x + y + z = 14
• xy + yz + zx = 7

To find:

x³ + y³ + z³ - 3xyz = ?

Solution:

Given that:

x + y + z = 14

Squaring on both sides,

(x + y + z)² = 14²

⇒ x² + y² + z² + 2(xy + yz + zx) = 196

⇒ x² + y² + z² + 2(7) = 196

⇒ x² + y² + z² = 182 ---- [Equation 1]

Now, for further solving this question apply a formula of algebra and substitute for the given values.

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

This can also be written as:

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

⇒ x³ + y³ + z³ - 3xyz = 14 (182 - 7)

⇒ x³ + y³ + z³ - 3xyz = 14 (175)

⇒ x³ + y³ + z³ - 3xyz = 2450

∴ x³ + y³ + z³ - 3xyz = 2450