4. On dividing ${x^3 – 3x^2 + x + 2}$ by a polynomial ${g(x)}$, the quotient and remainder
were ${x – 2}$ and ${–2x + 4}$, respectively. Find ${g(x)}$.
Solution:
Since, $p(x)= q(x)g(x)+r(x)$
given ; $q(x)$=${x – 2}$ and $r(x)$=${–2x + 4}$
Therefore $q(x)g(x)= p(x)-r(x)$
=${x^3 – 3x^2 + x + 2}$-$({–2x + 4})$
$q(x)g(x)$ = ${x^3-3x^2+3x-2}$
Now divide ${x^3-3x^2+3x-2}$ by the polynomial ${x-2}$
${x^3-3x^2+3x-2}$
${x-2}$
$x^2$
$-x$
$+1$
$x^3-2x^2$
($-$)
$(+)$
$-x^2+3x-2 $
-
$-x^2+2x$
($+$)
($-$)
${x-2}$
${x-2}$
($-$)($+$)
0
Answer:So, the required polynomial $g(x)$= $x^2 - x +1$