If $$ x^5 = 1 $$ , find the value of the fractional expression $$ \dfrac{x}{1+x^2}+\dfrac{x^2}{1+x^4}+\dfrac{x^3}{1+x}+\dfrac{x^4}{1+x^3} $$.

Question

If  $$ x^5 = 1 $$ , find the value of the fractional expression $$ \dfrac{x}{1+x^2}+\dfrac{x^2}{1+x^4}+\dfrac{x^3}{1+x}+\dfrac{x^4}{1+x^3} $$.

Uniteasy Admin · Accepted Answer

$$ \because x^5=1 $$
$$ 	herefore x^6 = x $$ , $$ x^7 = x^2 $$
$$ \dfrac{x}{1+x^2}+\dfrac{x^2}{1+x^4}+\dfrac{x^3}{1+x}+\dfrac{x^4}{1+x^3} $$
= $$ \dfrac{x}{1+x^2}\cdotp \dfrac{x^4}{x^4}  +\dfrac{x^2}{1+x^4}\cdotp \dfrac{x^3}{x^3}  +\dfrac{x^3}{1+x}\cdotp \dfrac{x^2}{x^2}  +\dfrac{x^4}{1+x^3}\cdotp \dfrac{x}{x}   $$
$$ =\dfrac{x^5}{x^4+x^6} +\dfrac{x^5}{x^3+x^7}+\dfrac{x^5}{x^2+x^3}+\dfrac{x^5}{x+x^4}   $$
$$ =\dfrac{1}{x^4+x}+\dfrac{1}{x^3+x^2}+\dfrac{1}{x^2+x^3}+\dfrac{1}{x+x^4} $$
$$ =\dfrac{2}{x}\cdotp\Big(  \dfrac{1}{1+x^3}+\dfrac{1}{x+x^2}\Big)    $$
$$ =\dfrac{2}{x}\cdotp\dfrac{x^3+x^2+x+1}{x+x^2+x^4+x^5} $$
$$ =\dfrac{2}{x}\cdotp\dfrac{x^3+x^2+x+1}{x^4+x^2+x+1} $$
$$ =2\cdotp \dfrac{x^3+x^2+x+1}{x^3+x^2+x+1} $$
$$ =2 $$

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