Give examples of polynomials $p (x), g (x), q (x)$ and $r (x)$ , which satisfy the division algorithm and $\deg r (x) = 0$

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Solution

Find the required polynomials:
Division algorithm is $D i v i d e n d = D i v i s o r \times Q u o t i e n t + R e m a i n d e r$
i.e., $p (x) = g (x) \times q (x)$ $+$ $r (x)$ , where $r (x) = 0$ or $\deg r (x) < \deg g (x)$
Let $p (x) = 5 x^{2} + 10 x + 20, g (x) = x^{2} + 2 x + 4$
Now, $q (x) = \frac{p (x)}{g (x)}$
$\Rightarrow$ $q (x) = 5$
As, $p (x) = g (x) \times q (x)$
So, $r (x) = 0$
Hence, the required polynomials are $p (x) = 5 x^{2} + 10 x + 20, g (x) = x^{2} + 2 x + 4$ , $q (x) = 5$ and $r (x) = 0$ .

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