Divide $(3 x^{3} + x^{2} + 2 x + 5)$ by $(x^{2} + 2 x + 1)$ , find the quotient and remainder.

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Solution

Step 1: Use long division method:
Let $p (x) = 3 x^{3} + x^{2} + 2 x + 5$ and $g (x) = x^{2} + 2 x + 1$ .
Let the quotient and remainder be $q (x) and r (x)$ .
Divide $p (x) by g (x),$ we get,
$\begin{matrix} x^{2} & + & 2 x & + & 1 \end{matrix} \begin{matrix} 3 x & - & 5 \end{matrix} \begin{matrix} 3 x^{3} & + & x^{2} & + & 2 x & + \begin{matrix} 5 \end{matrix} \\ 3 x^{3} & + & 6 x^{2} & + & 3 x \\ - & - & - \end{matrix} \begin{matrix} - & 5 x^{2} & - & x & + & 5 \\ - & 5 x^{2} & - & 10 x & - & 5 \\ + & + & + \end{matrix} \begin{matrix} 9 x & + & 10 \end{matrix}$
Therefore, $q (x) = (3 x - 5)$ and $r (x) = (9 x + 10)$ .
Step 2: Verify using division algorithm:
Formula:
$Dividend=Divisior×Quotient+Remainder$
By division algorithm, we have $p (x) = g (x) \times q (x) + r (x)$ .
$\Rightarrow (x^{2} + 2 x + 1) \times (3 x - 5) + (9 x + 10) = 3 x^{3} + 6 x^{2} + 3 x - 5 x^{2} - 10 x - 5 + 9 x + 10 = 3 x^{3} + x^{2} + 2 x + 5 = p (x)$
Thus, division algorithm is verified.
Hence, quotient is $q (x) = (3 x - 5)$ and the remainder is $r (x) = (9 x + 10)$ .

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