if X + Y + Z is equal to 8 and xy + Y Z plus ZX is equal to 20 find the value of x cube + y cube + Z cube minus 3 x y z
Answers
Answer:
⇒ ⇒ x³ + y³ + z³ - 3xyz = 80
Step-by-step explanation:
Given :
To find the value of : x³ + y³ + z³ - 3xyz ,
if x + y + z = 8,
xy + yz + zx = 20,.
Solution :
We know that,
(a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc
By substituting the values,
a = x , b = y , c = z
We get,
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2xz
⇒ (8)² = (x² + y² + z²) + 2(xy + yz + zx)
⇒ 64 = (x² + y² + z²) + 2(20)
⇒ 64 = (x² + y² + z²) + 40
⇒ 64 - 40 = (x² + y² + z²)
⇒ 24 = x² + y² + z² ...(i)
____
a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - ac - bc)
By substituting the values,
a = x , b = y , c = z
We get,
⇒ x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - xz)
⇒ x³ + y³ + z³ - 3xyz = (x + y + z)((x² + y² + z²) - (xy + yz + xz))
⇒ x³ + y³ + z³ - 3xyz = (20)(24 - 20)
⇒ x³ + y³ + z³ - 3xyz = 20 × 4
⇒ ⇒ x³ + y³ + z³ - 3xyz = 80
Step-by-step explanation:
answer is 32