Prove that: x2 + y2+ z2 - xy - yz - zx is always positive for distinct values - Maths - Polynomials

Question

Prove that: x2 + y2+ z2 - xy - yz
- zx is always positive for distinct values of x, y and z

Ashutosh Verma · Accepted Answer

Dear Student,

Please find below the solution to the asked query:

We have : x2 + y2 +
z2 - xy - yz -
zy

Now we multiply by in whole equation and get

2 ( x2 + y2 +
z2 - xy - yz - zx
)

⇒2 x2 + 2y2 +
2z2 - 2xy - 2yz -
2zx

⇒ x2 + x2 +
y2 + y2 +
z2 + z2 - 2xy -
2yz - 2zx

⇒ x2 + y2 -
2xy + y2 + z2 -
2yz + z2 + x2 -
2zx

⇒ ( x - y )2 + ( y - z
)2 + ( z - x )2

From above equation we can see that for distinct value of x ,
y and z given equation is always positive

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Prove that: x2 + y2+ z2 - xy - yz - zx is always positive for distinct values of x, y and z.