Polynomial Division (Long)

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Divide $3x^{3}-4x^{2}+4x-4\text{ by }3x-1$.

**Answer**

$\begin{array}{llllllllll}& | &x^{2}&-x&+1&\\& | &---- &---- &---- &\\3x-1& | & 3x^{3}&-4x^{2}&+4x&-4&\\& | &\pm\,3x^{3}&\mp\,x^{2}&\ &\\& | &---- &---- &---- &---- &\\& | &\ &-3x^{2}&+4x&-4&\\& | &\ &\mp\,3x^{2}&\pm\,x&\\& | &---- &---- &---- &---- &\\& | &\ &\ &3x&-4&\\& | &\ &\ &\pm\,3x&\mp\,1&\\& | &---- &---- &---- &---- &\\& | &\ &\ &\ &-3&\\\end{array}$

$\;\;\;3x^{3}-4x^{2}+4x-4$

$=(3x-1)(x^{2}-x+1)+(-3)$

$\text{or, }\frac{3x^{3}-4x^{2}+4x-4}{3x-1}=x^{2}-x+1+ \frac{-3}{3x-1}$