Find the quotient and remainder on dividing $p (x)$ by $g (x)$ in the following case, without actual division.
$p (x) = x^{3} + 4 x^{2} - 6 x + 2; g (x) = x - 3$

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Solution

To get the required remainder, equate $g (x)$ to $0$ and substitute that value of $x$ in $p (x)$ .

$x - 3 = 0 ⟹ x = 3$

Thus, $p (3)$ will be the required remainder.

$p (3) = 3^{3} + 4 (3)^{2} - 6 (3) + 2$
$= 27 + 36 - 18 + 2$
$= 47$

Hence value of the required remainder is $47$ .

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Divide the polynomial $p (x)$ by the polynomial $g (x)$ and find the quotient and remainder of the following.

$p (x) = x^{3} - 3 x^{2} + 5 x - 3$ and $g (x) = x^{2} - 2$ .

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