Polynomial Division (Long)

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Divide $12x^{3}+2x^{2}+5x+7\text{ by }3x+2$.

**Answer**

$\begin{array}{llllllllll}& | &4x^{2}&-2x&+3&\\& | &---- &---- &---- &\\3x+2& | & 12x^{3}&+2x^{2}&+5x&+7&\\& | &\pm\,12x^{3}&\pm\,8x^{2}&\ &\\& | &---- &---- &---- &---- &\\& | &\ &-6x^{2}&+5x&+7&\\& | &\ &\mp\,6x^{2}&\mp\,4x&\\& | &---- &---- &---- &---- &\\& | &\ &\ &9x&+7&\\& | &\ &\ &\pm\,9x&\pm\,6&\\& | &---- &---- &---- &---- &\\& | &\ &\ &\ &1&\\\end{array}$

$\;\;\;12x^{3}+2x^{2}+5x+7$

$=(3x+2)(4x^{2}-2x+3)+(1)$

$\text{or, }\frac{12x^{3}+2x^{2}+5x+7}{3x+2}=4x^{2}-2x+3+ \frac{1}{3x+2}$