Problems from Ruffini's rule

Do the division $\frac{p (x)}{q (x)}$ , where $p (x) = - x^{4} + a x^{3} - 3 x^{2} + 2 x - 3$ and $q (x) = x - 2$ , and impose the value of the parameter $a$ so that the division has a remainder equal to $3$ .

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Development:

We apply Ruffini's rule:

	$- 1$	$+ a$	$- 3$	$+ 2$	$- 3$
$2$		$- 2$	$2 (a - 2)$	$2 (2 (a - 2) - 3)$	$2 (2 (2 (a - 2) - 3) + 2)$
	$- 1$	$a - 2$	$2 (a - 2) - 3$	$2 (2 (a - 2) - 3) + 2$	$2 (2 (2 (a - 2) - 3) + 2) - 3$

Therefore, now we have to solve the following equation:

$2 (2 (2 (a - 2) - 3) + 2) - 3 = 3$

So:

$2 (2 (2 (a - 2) - 3) + 2) - 3 = 3 \Leftrightarrow 2 (2 (2 (a - 2) - 3) + 2) = 0 \Leftrightarrow$

$2 (2 (a - 2) - 3) + 2 = 0 \Leftrightarrow 2 (2 (a - 2) - 3) = - 2 \Leftrightarrow$

$2 (a - 2) - 3 = - 1 \Leftrightarrow 2 (a - 2) = 2 \Leftrightarrow a = 3$

Solution:

With the value of $a = 3$ , the result of the division has a remainder equal to $3$ .

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