Let x = y Let x = 0 Let a = b 0 x3 + y3 + z3 - 3xyz = x + y3 + z3 - 3xy(x + y + z) = x + y + z(x2 + y2 + z2 - xz - yz - 3xy) = x + y + z(x2 + y2 + z2 - xy - xz - yz)

1. 𝒑𝒈 𝟏𝟏𝟏 − 𝟏𝟏𝟐 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏

2. 𝒑𝒈 𝟏𝟏𝟏 − 𝟏𝟏𝟐 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏
𝑳𝒆𝒕 𝒙 = 𝒚
𝑳𝒆𝒕 𝒙 = 𝟎
𝑳𝒆𝒕 𝒙 = − 𝒚

3. 𝒑𝒈 𝟏𝟏𝟐 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏
𝑳𝒆𝒕 𝒂 = 𝒃
𝟎

4. 𝒑𝒈 𝟏𝟏𝟒 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏
𝒙𝟑 + 𝒚𝟑 + 𝒛𝟑 − 𝟑𝒙𝒚𝒛
= 𝒙𝟑
+ 𝒚𝟑
+ 𝟑𝒙𝟐
𝒚 + 𝟑𝒙𝒚𝟐
+ 𝒛𝟑
− 𝟑𝒙𝟐
𝒚 − 𝟑𝒙𝒚𝟐
− 𝟑𝒙𝒚𝒛
= 𝒙 + 𝒚 𝟑 + 𝒛𝟑 − 𝟑𝒙𝒚 𝒙 + 𝒚 + 𝒛
= 𝒙 + 𝒚 + 𝒛 𝒙 + 𝒚 𝟐
+ 𝒛𝟐
− 𝒙 + 𝒚 𝒛 − 𝟑𝒙𝒚 𝒙 + 𝒚 + 𝒛
= 𝒙 + 𝒚 + 𝒛 𝒙𝟐
+ 𝒚𝟐
+ 𝟐𝒙𝒚 + 𝒛𝟐
− 𝒙𝒛 − 𝒚𝒛 − 𝟑𝒙𝒚
= 𝒙 + 𝒚 + 𝒛 𝒙𝟐
+ 𝒚𝟐
+ 𝒛𝟐
− 𝒙𝒚 − 𝒙𝒛 − 𝒚𝒛

5. 𝒑𝒈 𝟏𝟏𝟐 − 𝟏𝟏𝟑 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏

6. 𝒑𝒈 𝟏𝟏𝟑 − 𝟏𝟏𝟒 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏

7. 𝒑𝒈 𝟏𝟏𝟒 𝒄𝒚𝒄𝒍𝒊𝒄 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏