If the polynomial f=x^4+x^2+1 is divided by g=x^2+2x+3 what is the reminder of division ?

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We have to  divide f = x^4 + x^2 + 1 by g=x^2 + 2x + 3.

 

x^2 + 2x + 3 | x^4 + x^2 + 1.................| x^2 -2x + 2

.....................  x^4 + 2x^3 + 3x^2

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................................2x^3 - 2x^2 + 1

................................2x^3 - 4x^2 - 6x

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.......................................2x^2 + 6x + 1

.......................................2x^2 + 4x + 6

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....................................................2x - 5

Therefore we get the remainder as 2x - 5

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Given the polynomial f(x) = x^4 + x^2 + 1

Divided by g(x) = x^2 + 2x + 3.

We need to find the remainder.

First we will divide and determine the quotient.

==> f(x) = g(x)*P(x) + R where R is the remainder

==> f(x) = g(x) * (x^2 -2x +2) + (2x-5)

Then the quotient is ( x^2 - 2x +2)

And the remainder is ( 2x -5).

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