Question-24

Consider: \[\begin{equation*} f( x) =\frac{x^{4} +x^{2} +1}{x^{2} +x+1} \end{equation*}\] If \(\displaystyle f^{\prime }( x) =ax+b\), find \(\displaystyle a+b\).

Hint

Polynomial long division

Answer

\(1\)

Solution

Using the identity:

\[\begin{equation*} x^{4} +x^{2} +1=\left( x^{2} +x+1\right)\left( x^{2} -x+1\right) \end{equation*}\] we have:

\[\begin{equation*} f( x) =x^{2} -x+1 \end{equation*}\]

From this, we get \(\displaystyle f^{\prime }( x) =2x-1\). Thus \(\displaystyle a=2,b=-1\) and \(\displaystyle a+b=1\).