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

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Solution

linear division method

$(x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5) \div (x + 2)$

$= x^{3} (x + 2) - 2 x^{3} + 2 x^{3} + 3 x^{2} + 4 x + 5$

$= x^{3} (x + 2) + 3 x^{2} + 4 x + 5$

$= x^{3} (x + 2) + 3 x (x + 2) - 6 x + 4 x + 5$

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

$= x^{3} (x + 2) + 3 x (x + 2) - 2 (x + 2) + 4 + 5$

$= x^{3} (x + 2) + 3 x (x + 2) - 2 (x + 2) + 9$

$= (x + 2) (x^{3} + 3 x - 2) + 9$

quotient $= x^{3} + 3 x - 2$

remainder $= 9$

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p (x) = x^{4} - 3 x^{2} + 4 x + 5

g (x) = x^{2} + 1 - x

p (x)

g (x)

p (x) = x^{4} - 3 x^{2} + 4 x + 5

g (x) = x^{2} + 1 - x

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