Divide $30 x^{4} + 11 x^{3} - 82 x^{2} + 48 - 12 x$ by $3 x^{2} + 2 x - 4$ and check the result by using division algorithm.

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Solution

We have to divide $30 x^{4} + 11 x^{3} - 82 x^{2} - 12 x + 48$ by $3 x^{2} + 2 x - 4$

$10 x^{2} - 3 x - 12$
$3 x^{2} + 2 x - 4) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 30 x^{4} + 11 x^{3} - 82 x^{2} - 12 x + 48$
$30 x^{4} + 20 x^{3} - 40 x^{2}$
$(-)$ $(-)$ $(+)$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 9 x^{3} - 42 x^{2} - 12 x + 48$
$- 9 x^{3} - 6 x^{2} + 12 x$
$(+)$ $(+)$ $(-)$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ - 36 x^{2} - 24 x + 48$
$- 36 x^{2} - 24 x + 48$
$(+)$ $(+)$ $(-)$
$¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ 0$

Now, to verify, use the formula,
Dividend $=$ Divisor $\times$ Quotient $+$ Reminder
$= (3 x^{2} + 2 x - 4) \times (10 x^{2} - 3 x - 12) + 0$
$= 30 x^{4} - 9 x^{3} - 36 x^{2} + 20 x^{3} - 6 x^{2} - 24 x - 40 x^{2} + 12 x + 48$
$= 30 x^{4} + 11 x^{3} - 82 x^{2} - 12 x + 48$

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