# How do you divide (-x^4-4x^3-9x^2-7x-7)/(x-2) ?

##### 1 Answer

$- {x}^{3} - 6 {x}^{2} - 21 x - 49$ with a remainder of $- 105$

#### Explanation:

We can do long division on this:

$\textcolor{w h i t e}{\frac{x - 2}{\textcolor{b l a c k}{x - 2}} \frac{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\text{)} \overline{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}}}}$

We multiply $x$ by $- {x}^{3}$ to get to $- {x}^{4}$, and so we have:

$\textcolor{w h i t e}{\frac{x - 2}{\textcolor{b l a c k}{x - 2}} \frac{- {x}^{4} \textcolor{b l a c k}{\textcolor{w h i t e}{00} - {x}^{3}} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\text{)} \overline{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}}}}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{- {x}^{4} + 2 {x}^{3}} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\overline{0 {x}^{4} - 6 {x}^{3} - 9 {x}^{2}}} - 7 x - 7}\right)}$

We multiply $x$ by $- 6 {x}^{2}$ to get to $- 6 {x}^{3}$ and so we have:

$\textcolor{w h i t e}{\frac{x - 2}{\textcolor{b l a c k}{x - 2}} \frac{- {x}^{4} \textcolor{b l a c k}{\textcolor{w h i t e}{00} - {x}^{3} - 6 {x}^{2}} - 7 x - 7}{\textcolor{b l a c k}{\text{)} \overline{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}}}}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{- {x}^{4} + 2 {x}^{3}} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\overline{0 {x}^{4} - 6 {x}^{3} - 9 {x}^{2}}} - 7 x - 7}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4}} - 6 {x}^{3} + 12 {x}^{2}} - 7 x - 7}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} -} 0 {x}^{3} - 21 {x}^{2} - 7 x}} - 7}\right)}$

We multiply $x$ by $- 21 x$ to get to $- 21 {x}^{2}$ and so we have:

$\textcolor{w h i t e}{\frac{x - 2}{\textcolor{b l a c k}{x - 2}} \frac{- {x}^{4} \textcolor{b l a c k}{\textcolor{w h i t e}{00} - {x}^{3} - 6 {x}^{2} - 21 x} - 7}{\textcolor{b l a c k}{\text{)} \overline{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}}}}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{- {x}^{4} + 2 {x}^{3}} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\overline{0 {x}^{4} - 6 {x}^{3} - 9 {x}^{2}}} - 7 x - 7}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4}} - 6 {x}^{3} + 12 {x}^{2}} - 7 x - 7}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} -} 0 {x}^{3} - 21 {x}^{2} - 7 x}} - 7}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4} - 2 {x}^{3}} - 21 {x}^{2} + 42 x} - 7}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} - 0 {x}^{3}} - 0 {x}^{2} - 49 x - 7}}}\right)}$

We multiply $x$ by $- 49$ to get to $- 49 x$ and so we have:

$\textcolor{w h i t e}{\frac{x - 2}{\textcolor{b l a c k}{x - 2}} \frac{- {x}^{4} \textcolor{b l a c k}{\textcolor{w h i t e}{00} - {x}^{3} - 6 {x}^{2} - 21 x - 49}}{\textcolor{b l a c k}{\text{)} \overline{- {x}^{4} - 4 {x}^{3} - 9 {x}^{2} - 7 x - 7}}}}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{- {x}^{4} + 2 {x}^{3}} - 9 {x}^{2} - 7 x - 7}{\textcolor{b l a c k}{\overline{0 {x}^{4} - 6 {x}^{3} - 9 {x}^{2}}} - 7 x - 7}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4}} - 6 {x}^{3} + 12 {x}^{2}} - 7 x - 7}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} -} 0 {x}^{3} - 21 {x}^{2} - 7 x}} - 7}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4} - 2 {x}^{3}} - 21 {x}^{2} + 42 x} - 7}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} - 0 {x}^{3}} - 0 {x}^{2} - 49 x - 7}}}\right)}$
$\textcolor{w h i t e}{\frac{x - 2}{x - 2} \left(\frac{\textcolor{b l a c k}{\textcolor{w h i t e}{- {x}^{4} - 2 {x}^{3} - 1 {x}^{2}} - 49 x + 98}}{\textcolor{b l a c k}{\overline{\textcolor{w h i t e}{0 {x}^{4} - 0 {x}^{3} - 0 {x}^{2}} - 0 x - 105}}}\right)}$

And so $\frac{- {x}^{4} - {4}^{3} - 9 {x}^{2} - 7 x - 7}{x - 2} = - {x}^{3} - 6 {x}^{2} - 21 x - 49$ with a remainder of $- 105$