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  4. Let P ( x )= x 2- cx - c. If P ( x ) is divided by ( x -2), the remainder is the same as when ( P ( x ))2 is divided by (x-2), then the sum of all possible values of c, is

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Q. Let $P ( x )= x ^2- cx - c$. If $P ( x )$ is divided by $( x -2)$, the remainder is the same as when $( P ( x ))^2$ is divided by $(x-2)$, then the sum of all possible values of $c$, is

Complex Numbers and Quadratic Equations

Solution:

$P(x)=(x-2) Q_1+R_1 $
$(P(x))^2=(x-2) Q_2+R_2 $
$\text { Given, } R_1=R_2$
Given, $R_1=R_2$.
$\Rightarrow P (2)=( P (2))^2 \Rightarrow P (2)=0$ or 1
Hence, $4-3 c =0$ or 1
$\therefore c =\frac{4}{3}$ or $1 \Rightarrow$ Sum $=\frac{7}{3}$.