What is  the reminder when f(x)=x^5+x^4+1 is divided by g(x)=(x-1)^3 ?

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We have f(x) = x^5+x^4+1 and g(x) = (x- 1)^3 = x^3 - 3x^2 + 3x - 1.

We can find the remainder by dividing them.

x^3 - 3x^2 + 3x - 1 | x^5+x^4+1                      |x^2 + 4x + 9

................................x^5 - 3x^4 + 3x^3 - x^2

-----------------------------------------------------------

.................................0 +...

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We have f(x) = x^5+x^4+1 and g(x) = (x- 1)^3 = x^3 - 3x^2 + 3x - 1.

We can find the remainder by dividing them.

x^3 - 3x^2 + 3x - 1 | x^5+x^4+1                      |x^2 + 4x + 9

................................x^5 - 3x^4 + 3x^3 - x^2

-----------------------------------------------------------

.................................0 + 4x^4 - 3x^3 + x^2 +1

.......................................4x^2 - 12x^3 + 12x^2 - 4x

----------------------------------------------------------------

......................................0 + 9x^3 - 11x^2 + 4x + 1

............................................9x^3 - 27x^2 + 27x - 9

---------------------------------------------------------------

.............................................0 +  16x^2 - 23x + 10

Therefore the remainder is 16x^2 - 23x +10.

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