Question

Verify the cyclic symmetry of the following expressions if a, b and c are the variables:

(1) a(b − c) + b(c − a) + c(a − b)

(2) ab(a + b) + bc(b + c) + ca(c + a)

(3) a²b(a − b) + b²c(b − c) + c²a(c − a)

(4) a²(a + b) + b²(b + c) + c²(c + a)

(5) ab²(a − b) + bc²(b − c) + ca²(c − a)

Answer 1

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

Changing the variables 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 a, b and c 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 a, b and c 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 a, b and c 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 a, b and c cyclically, we get ... (2)

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

Hence, the expression is a cyclic symmetrical expression.