Divide each of the following polynomial by synthetic method and also by linear division method. Write quotient and the remainder.
$(x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5) \div (x + 2)$

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Solution

Synthetic Division :
Divided = $x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5$
Divisor $= x + 2$
Opposite of $2 = - 2$
The coefficient from of the quotient is $(1, 0, 3, - 2)$
$∴ Q u o t i e n t = x^{3} + 3 x - 2$ and remainder $= 9$
Linear method
$x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5$
$x^{3} (x + 2 + 3 x (x + 2) - 6 x + 4 x + 5$
$x^{3} (x + 2) + 3 x (x + 2) - 2 x + 5$
$x^{3} (x + 2) + 3 x (x + 2) - 2 (x + 2) + 4 + 5$
$(x + 2) \times (x^{3} + 3 x - 2) + 9$

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