The value of frac{1}{a^{2}+a x+x^{2}}-frac{1} | vedclass

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(D) Let the expression be $E = \frac{1}{a^{2}+a x+x^{2}}-\frac{1}{a^{2}-a x+x^{2}}+\frac{2 a x}{a^{4}+a^{2} x^{2}+x^{4}}$.First,simplify the first two

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(D) Let the expression be $E = \frac{1}{a^{2}+a x+x^{2}}-\frac{1}{a^{2}-a x+x^{2}}+\frac{2 a x}{a^{4}+a^{2} x^{2}+x^{4}}$.First,simplify the first two terms by taking the common denominator $(a^{2}+a x+x^{2})(a^{2}-a x+x^{2})$.Note that $(a^{2}+a x+x^{2})(a^{2}-a x+x^{2}) = (a^{2}+x^{2})^{2} - (ax)^{2} = a^{4} + 2a^{2}x^{2} + x^{4} - a^{2}x^{2} = a^{4} + a^{2}x^{2} + x^{4}$.So,the expression becomes:$E = \frac{(a^{2}-a x+x^{2}) - (a^{2}+a x+x^{2})}{a^{4}+a^{2}x^{2}+x^{4}} + \frac{2 a x}{a^{4}+a^{2} x^{2}+x^{4}}$$E = \frac{a^{2}-a x+x^{2}-a^{2}-a x-x^{2}}{a^{4}+a^{2}x^{2}+x^{4}} + \frac{2 a x}{a^{4}+a^{2} x^{2}+x^{4}}$$E = \frac{-2ax}{a^{4}+a^{2}x^{2}+x^{4}} + \frac{2ax}{a^{4}+a^{2} x^{2}+x^{4}}$$E = 0$.

The value of $\frac{1}{a^{2}+a x+x^{2}}-\frac{1}{a^{2}-a x+x^{2}}+\frac{2 a x}{a^{4}+a^{2} x^{2}+x^{4}}$ is

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