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  4. If (d/d x)((1+x2+x4/1+x+x2))=a x +b, then (a, b)=

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Q. If $\frac{d}{d x}\left(\frac{1+x^{2}+x^{4}}{1+x+x^{2}}\right)=a x +b$, then $(a, b)=$

Solution:

$a x +b=\frac{d}{d x}\left\{\frac{1+x^{2}+x^{4}}{1+x+x^{2}}\right\}$
$=\frac{d}{d x}\left\{\frac{\left(1+x+x^{2}\right)\left(1-x+x^{2}\right)}{1+x+x^{2}}\right\}$
$=\frac{d}{d x}\left\{1-x+x^{2}\right\} =-1+2 x$
$\Rightarrow a=2, b=-1 .$