If x+y+z = 8 and xy+yz+zx =20, find the value of x³ +y³ +z³ -3xyz.
Answers
Given : x + y + z = 8 and xy + yz + zx = 20
On Squaring, x + y + z = 8 both sides, we get
(x + y + z)² = 8²
x² + y² + z² + 2 (xy + yz + zx) = 64
[By using an identity, (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx)]
x² + y² + z² + 2 x 20 = 64
x² + y² + z² + 40 = 64
x² + y² + z² = 64 - 40
x² + y² + z² = 24………..(1)
Now, by using an identity , x³ + y³ + z³ - 3xyz= (x + y + z) [(x² + y² + z² - xy - yz - zx)]
x³ + y³ + z³ - 3xyz = (x + y + z) [(x² + y² + z² - (xy + yz + zx)]
= 8[24 – 20]
[Given : x + y + z = 8 and xy + yz + zx = 20 and from eq1]
= 8 x 4
= 32
x³ + y³ + z³ - 3xyz = 32
Hence the value of x³ + y³ + z³ - 3xyz is 32.
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Answer:
Step-by-step explanation:
Given : x + y + z = 8 and xy + yz + zx = 20
On Squaring, x + y + z = 8 both sides, we get
(x + y + z)² = 8²
x² + y² + z² + 2 (xy + yz + zx) = 64
[By using an identity, (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx)]
x² + y² + z² + 2 x 20 = 64
x² + y² + z² + 40 = 64
x² + y² + z² = 64 - 40
x² + y² + z² = 24………..(1)
Now, by using an identity , x³ + y³ + z³ - 3xyz= (x + y + z) [(x² + y² + z² - xy - yz - zx)]
x³ + y³ + z³ - 3xyz = (x + y + z) [(x² + y² + z² - (xy + yz + zx)]
= 8[24 – 20]
[Given : x + y + z = 8 and xy + yz + zx = 20 and from eq1]
= 8 x 4
= 32
x³ + y³ + z³ - 3xyz = 32