Divide the polynomial $p (x)$ by the polynomial $g (x)$ and find quotient and remainder for the following $p (x) = 8 x^{3} - 6 x + 7; g (x) = 2 x - 1$

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Solution

Step 1: Find the quotient and remainder by long division method:
As, $p (x) = 8 x^{3} - 6 x + 7$
$g (x) = 2 x - 1$
Let the quotient and remainder be $q (x)$ and $r (x)$
$\begin{matrix} 2 x - & 1 \end{matrix} \begin{matrix} 4 x^{2} & + & 2 x & - & 2 \end{matrix} \begin{matrix} \end{matrix} \begin{matrix} 8 x^{3} & + 0 x^{2} & - & 6 x & + & 7 \end{matrix} \begin{matrix} 8 x^{3} & - & 4 x^{2} \\ - & + \end{matrix} \begin{matrix} 4 x^{2} & - & 6 x & + & 7 \\ 4 x^{2} & - & 2 x \\ - & + \end{matrix} \begin{matrix} - 4 x & + & 7 \\ - 4 x & + & 2 \\ + & - \end{matrix} \begin{matrix} + & 5 \end{matrix}$
Clearly, the quotient $= q (x) = \begin{matrix} 4 x^{2} & + & 2 x & - & 2 \end{matrix}$
and the remainder $= r (x) = 5$
Step 2: Verify the division algorithm:
If $f (x)$ and $g (x)$ are any two polynomials with $g (x) \neq 0, then f (x) = g (x) . q (x) + r (x),$
where , $r(x)=0ordegr(x)<degg(x).$
$q (x) \times g (x) + r (x) = (2 x - 1) (4 x^{2} + 2 x - 2) + 5 = 8 x^{3} + 0 x^{2} - 6 x + 2 + 5 = 8 x^{3} - 6 x + 7 = p (x)$
Hence, the division algorithm is verified.

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