If $\Delta=\left|\begin{array}{ccc}1+x & x & x^2 \ x & 1+x & x^2 \ x^2 & x & 1+x\end{array} ight|

Question

If $\Delta=\left|\begin{array}{ccc}1+x & x & x^2 \ x & 1+x & x^2 \ x^2 & x & 1+x\end{array}ight|=a x^5+b x^4+c x^3+d x^2+e x+f$

Professor PC Thomas Classes · Answer

Put $x=0 \Rightarrow f=\Delta(0)=1$ $ e=\Delta^{\prime}(0)=3 $ Put $x=1$ $$ \left|\begin{array}{lll} 2 & 1 & 1 \ 1 & 2 & 1 \ 1 & 1 & 2 \end{array}ight|=a+b+c+d+e+f $$ $$ \begin{equation*} 	herefore a+b+c+d=0 	ag{i} \end{equation*} $$ Put $x=-1$ $$ \begin{align*} \Rightarrow & -a+b-c+d-3+1=0 \ & -a+b-c+d=2 \ldots(	ext { ii }) 	ag{ii} \end{align*} $$ On solving equation (i) & (ii), we get $b+d=1$ and $a+c=-1$