Question

For what value of k, (x + y + z)2 + k(x2 + y2 + z2) can be resolved into linear rational factors?​

Answer 1

Given :  (x + y + z)2 + k(x2 + y2 + z2) can be resolved into linear rational factors

To Find :  Value of k  

Solution:

(x + y + z)² + k(x² + y²+ z²)

Expanding (x + y + z)²  using x² + y²+ z² + 2(xy + yz + xz)

=  x² + y²+ z² + 2(xy + yz + xz) + k(x² + y²+ z²)

= (x² + y²+ z²)(1 + k) + 2(xy + yz + xz)

linear rational factors

Hence x² + y²+ z² coefficient must be zero

=> 1 + k must be zero

1 + k = 0

=> k = - 1

Hence Value of k is - 1

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