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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(xy) + yz(yz) + zx(zx)

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

(4) x3(x + y) + y3(y + z) + z3(z + x)

(5) xy2(xy) + yz2(yz) + zx2(zx)


Solution

(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


Mathematics
Mathematics
Standard X

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