Simplify : $\frac{3x+1}{9x^2+3x+1} + \frac{3x-1}{9x^2-3x+1} + \frac{2}{81x^4+9x^2+1}$
तपाईं प्रतिभाशाली हुनुहुन्छ । यस्ता समस्या त कति समाधान गर्नु भयो कति, हाेइन र?
तपाईं अफुलाई गणितमा अब्बल बनाउन चहानुहुन्छ भने यस प्रश्नमा दिइएको समस्याकाे समाधान गरेर देखाउनुहाेस् ।
If you want to be a genius in Mathematics, solve the followint problem.
Genius can solve any mathematical problem.
Mathematics Injection
$\large \textbf{Simplify:}$
Simplify : $\frac{3x+1}{9x^2+3x+1} + \frac{3x-1}{9x^2-3x+1} + \frac{2}{81x^4+9x^2+1}$
Here,
$\frac{3x+1}{9x^2+3x+1} + \frac{3x-1}{9x^2-3x+1} + \frac{2}{81x^4+9x^2+1}$
= $\frac{(3x+1)(9x^2-3x+1) + (3x-1)(9x^2+3x+1)}{(9x^2+3x+1)(9x^2-3x+1)}+ \frac{2}{81x^4+9x^2+1} $
= $\frac{(3x)^3 + 1^3 + (3x)^3 - 1^3}{(9x^2)^2+(3x)^2.1^2+(1^2)^2}+ \frac{2}{81x^4+9x^2+1} $
= $\frac{27x^3 + 27x^3}{81x^4+9x^2+1}+ \frac{2}{81x^4+9x^2+1}$
= $\frac{54x^3}{81x^4+9x^2+1}+ \frac{2}{81x^4+9x^2+1}$
= $\frac{54x^3+2}{81x^4+9x^2+1}$
= $\frac{2(27x^3+1)}{81x^4+9x^2+1}$
= $\frac{2\{(3x)^3+1^3\}}{81x^4+9x^2+1}$
= $\frac{2(3x+1)(9x^2-3x+1)}{(9x^2+3x+1)(9x^2-3x+1)}$
= $\frac{2(3x+1)}{(9x^2+3x+1)}.$ Ans
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