Question

If x, y and z are variables, verify the cyclic symmetry of the following expressions.

(1) x(y + z) + y(z + x) + z(x + y)

(2) xy(x − y) + yz(y − z) + zx(z − x)

(3) x²y(x + y) + y²z(y + z) + z²x(z + x)

(4) x³(x + y) + y³(y + z) + z³(z + x)

(5) xy²(x − y) + yz²(y − z) + zx²(z − x)

Answer 1

(1) The given expression is ... (1)

Changing the variables x, y and z cyclically, we get ... (2)

Comparing expressions (1) and (2), we observe that they are same.

Hence, the expression is a cyclic symmetrical expression

(2) The given expression is ... (1)

Changing the variables x, y and z cyclically, we get ... (2)

Comparing expressions (1) and (2), we observe that they are same.

Hence, the expression is a cyclic symmetrical expression

(3) The given expression is ... (1)

Changing the variables x, y and z cyclically, we get ... (2)

Comparing expressions (1) and (2), we observe that they are same.

Hence, the expression is a cyclic symmetrical expression.

(4) The given expression is ... (1)

Changing the variables x, y and z cyclically, we get ... (2)

Comparing expressions (1) and (2), we observe that they are same.

Hence, the expression is a cyclic symmetrical expression

(5) The given expression is ... (1)

Changing the variables x, y and z cyclically, we get ... (2)

Comparing expressions (1) and (2), we observe that they are same.

Hence, the expression is a cyclic symmetrical expression