Math, asked by Abdul5497, 10 months ago

If x+y+z = 8 and xy+yz+zx =20, find the value of x³ +y³ +z³ -3xyz.

Answers

Answered by nikitasingh79
7

 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.

HOPE THIS ANSWER WILL HELP YOU…..

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Answered by Anonymous
3

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

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