A) \[(a+b)(a-b)\]
B) \[(a+b)(a+b)\]
C) \[(a-b)(a-b)\]
D) \[(a-b)(a-b)\]
Correct Answer: A
Solution :
| [a] \[{{({{a}^{2}}-{{b}^{2}})}^{3}}+{{({{b}^{2}}-{{c}^{2}})}^{3}}+{{({{c}^{2}}-{{a}^{2}})}^{3}}\] \[={{a}^{6}}-{{b}^{6}}-3{{a}^{4}}{{b}^{2}}+3{{a}^{2}}{{b}^{4}}+{{b}^{6}}-{{c}^{6}}\] \[-\,3{{b}^{4}}{{c}^{2}}+3{{b}^{2}}{{c}^{4}}+{{c}^{6}}-{{a}^{6}}-\,3{{c}^{4}}{{a}^{2}}+3{{c}^{2}}{{a}^{4}}\] \[=3\,[{{a}^{2}}{{b}^{4}}-{{a}^{4}}{{b}^{2}}+{{b}^{2}}{{c}^{4}}-{{b}^{4}}{{c}^{2}}+\,{{c}^{2}}{{a}^{4}}-{{c}^{4}}{{a}^{2}}]\] \[=3\,[{{a}^{2}}{{b}^{2}}({{b}^{2}}-{{a}^{2}})+{{c}^{4}}({{b}^{2}}-{{a}^{2}})+\,{{c}^{2}}({{a}^{4}}-{{b}^{4}})]\] \[=3\,({{b}^{2}}-{{a}^{2}})[{{a}^{2}}{{b}^{2}}+{{c}^{4}}-{{c}^{2}}{{a}^{2}}+{{c}^{2}}{{b}^{2}}]\] \[=-3\,({{a}^{2}}-{{b}^{2}})[{{a}^{2}}{{b}^{2}}+{{c}^{4}}-{{c}^{2}}{{a}^{2}}-{{c}^{2}}{{b}^{2}}]\] \[=-3\,({{a}^{2}}-{{b}^{2}})[{{a}^{2}}({{b}^{2}}-{{c}^{2}})-{{c}^{2}}({{b}^{2}}-{{c}^{2}})]\] \[=-\,3\,({{a}^{2}}-{{b}^{2}})({{b}^{2}}-{{c}^{2}})({{a}^{2}}-{{c}^{2}})\] \[=3\,({{a}^{2}}-{{b}^{2}})({{b}^{2}}-{{c}^{2}})({{c}^{2}}-{{a}^{2}})\] Thus, \[({{a}^{2}}-{{b}^{2}})\]i.e., \[(a+b)(a-b)\]is the required factor. |
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