Polynomial Division (Long)

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Divide $9x^{3}-12x^{2}-4x+5\text{ by }3x-2$.

**Answer**

$\begin{array}{llllllllll}& | &3x^{2}&-2x&-3&\\& | &---- &---- &---- &\\3x-2& | & 9x^{3}&-12x^{2}&-4x&+5&\\& | &\pm\,9x^{3}&\mp\,6x^{2}&\ &\\& | &---- &---- &---- &---- &\\& | &\ &-6x^{2}&-4x&+5&\\& | &\ &\mp\,6x^{2}&\pm\,4x&\\& | &---- &---- &---- &---- &\\& | &\ &\ &-8x&+5&\\& | &\ &\ &\mp\,8x&\pm\,\frac{16}{3}&\\& | &---- &---- &---- &---- &\\& | &\ &\ &\ &-\frac{1}{3}&\\\end{array}$

$\;\;\;9x^{3}-12x^{2}-4x+5$

$=(3x-2)(3x^{2}-2x-\frac{8}{3})+(-\frac{1}{3})$

$\text{or, }\frac{9x^{3}-12x^{2}-4x+5}{3x-2}=3x^{2}-2x-\frac{8}{3}+ \frac{-\frac{1}{3}}{3x-2}$