Divide $p (x)$ by $g (x)$ in the following cases and verify division algorithm.
$p (x) = x^{4} - 4 x^{2} + 12 x + 9; g (x) = x^{2} + 2 x - 3$

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Solution

The division of $p (x) = x^{4} - 4 x^{2} + 12 x + 9$ by $g (x) = x^{2} + 2 x - 3$ is as shown above:

Now we know that the division algorithm states that:
Dividend $= ($ Divisor $\times$ quotient $) +$ Remainder

Here, the dividend is $x^{4} - 4 x^{2} + 12 x + 9$ , the divisor is $x^{2} + 2 x - 3$ , the quotient is $x^{2} - 2 x + 3$ and the remainder is $18$ , therefore,

$x^{4} - 4 x^{2} + 12 x + 9 = [(x^{2} + 2 x - 3) (x^{2} - 2 x + 3)] + 18 \Rightarrow x^{4} - 4 x^{2} + 12 x + 9 = [x^{2} (x^{2} - 2 x + 3) + 2 x (x^{2} - 2 x + 3) - 3 (x^{2} - 2 x + 3)] + 18 \Rightarrow x^{4} - 4 x^{2} + 12 x + 9 = (x^{4} - 2 x^{3} + 3 x^{2} + 2 x^{3} - 4 x^{2} + 6 x - 3 x^{2} + 6 x - 9) + 18 \Rightarrow x^{4} - 4 x^{2} + 12 x + 9 = x^{4} - 2 x^{3} + 2 x^{3} + 3 x^{2} - 4 x^{2} - 3 x^{2} + 6 x + 6 x - 9 + 18 \Rightarrow x^{4} - 4 x^{2} + 12 x + 9 = x^{4} - 4 x^{2} + 12 x + 9$

Hence, the division algorithm is verified.

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