Verify division algorithm for the polynomials $f (x) = 8 + 20 x + x^{2} - 6 x^{3}$ and $g (x) = 2 + 5 x - 3 x^{2} .$

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Solution

First, we write the given polynomials in standard form in decreasing order of degree and then perform the division as shown below.

$f (x) = 8 + 20 x + x^{2} - 6 x^{3}$
$f (x) = - 6 x^{3} + x^{2} + 20 x + 8$

$g (x) = 2 + 5 x - 3 x^{2}$
$g (x) = - 3 x^{2} + 5 x + 2$
So

$\frac{f (x)}{g (x)}$

$r (x) = x + 8$

$q (x) = 2 x + 3$

$\Rightarrow (Q u o t i e n t \times d i v i s o r) + r e m a i n d e r$

$= (2 x + 3) (- 3 x^{2} + 5 x + 2) + x + 2$

$= - 6 x^{3} + 10 x^{2} + 4 x - 9 x^{2} + 15 x + 6 + x + 2$

Therefore, $- 6 x^{3} + x^{2} + 20 x + 8 = d i v i d e n d$

Thus, $(Q u o t i e n t \times d i v i s o r) + r e m a i n d e r = d i v i d e n d$

Hence, the division algorithm is verified

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Verify division algorithm for the polynomials $f (x) = 8 + 20 x + x^{2} - 6 x^{3}$ and $g (x) = 2 + 5 x - 3 x^{2} .$

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