Value of \( \Large \frac{ \left(a^{2}-b^{2}\right)^{3}+ \left(b^{2}-c^{2}\right)^{3}+ \left(c^{2}-a^{2}\right)^{3} }{ \left(a-b\right)^{3}+ \left(b-c\right)^{3}+ \left(c-a\right)^{3} } \)
Correct Answer: |
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| D) \( \Large \left(a+b\right)+ \left(b+c\right)+ \left(c+a\right) \) |
Description for Correct answer:
As we know that,
If a + b + c = 0,
then \( \Large a^{3}+b^{3}+c^{3}=3abc \)
Therefore, \( \Large \left(a^{2}-b^{2}\right)^{3}+ \left(b^{2}-c^{2}\right)^{3}+ \left(c^{2}-a^{2}\right)^{3}=3 \left(a^{2}-b^{2}\right) \left(b^{2}-c^{2}\right) \left(c^{2}-a^{2}\right) \)
and \( \Large \left(a-b\right)^{3}+ \left(b-c\right)^{3}+ \left(c-a\right)^{3}=3 \left(a-b\right) \left(b-c\right) \left(c-a\right) \)
Therefore, \( \Large \frac{ \left(a^{2}-b^{2}\right)^{3}+ \left(b^{2}-c^{2}\right)^{3}+ \left(c^{2}-a^{2}\right)^{3} }{ \left(a-b\right)^{3}+ \left(b-c\right)^{3}+ \left(c-a\right)^{3} } \)
= \( \Large \frac{3 \left(a^{2}-b^{2}\right) \left(b^{2}-c^{2}\right) \left(c^{2}-a^{2}\right) }{3 \left(a-b\right) \left(b-c\right) \left(c-a\right) } \)
= \( \Large \frac{ \left(a-b\right) \left(a+b\right) \left(b-c\right) \left(b+c\right) \left(c-a\right) \left(c+a\right) }{ \left(a-b\right) \left(b-c\right) \left(c-a\right) } \)
= \( \Large \left(a+b\right) \left(b+c\right) \left(c+a\right) \)
Part of solved Rational expression questions and answers :
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