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# $\large\frac{x^2+x+1}{(x-1)(x-2)(x-3)}=\frac{A}{x-1}+\frac{B}{x-2}+\frac{C}{x-3}$ then find $A+C=$

$\begin {array} {1 1} (a)\;4 & \quad (b)\;5 \\ (c)\;6 & \quad (d)\;8 \end {array}$

<div class="clay6-step-odd"><div class="clay6-basic" id="pr10">Expand the equation and write it in matrix form. You will end up with -</div><div class="clay6-basic" id="pr11">$\begin{bmatrix} A \\ B \\ C \end{bmatrix} = {\begin{bmatrix} 1 &amp; 1 &amp; 1 \\ -5 &amp; -4 &amp; -3 \\ 6 &amp; 3 &amp; 2 \end{bmatrix} }^{\Large -1} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$</div><div class="clay6-basic" id="pr12">The right hand side when solved will end up with</div><div class="clay6-basic" id="pr13">$A = \frac{3}{2} \; B = 3 \; C = \frac{13}{2}$</div><div class="clay6-basic" id="pr14">So $A + C = 8$. Hence (d) is the right answer.</div></div>
edited Feb 3, 2014 by pady_1