# How do you use synthetic division to divide (180x-x^4)/(x-6)?

Sep 2, 2017

$\frac{180 x - {x}^{4}}{x - 6} = - {x}^{3} - 6 {x}^{2} - 36 x - 36 - \frac{216}{x - 6}$

#### Explanation:

Write $180 x - {x}^{4}$ in standard form, with the ${x}^{4}$ term first and 0 as the coefficient for the "missing" terms: $- {x}^{4} + 0 {x}^{3} + 0 {x}^{2} + 180 x + 0$ (The last zero is for the constant. For synthetic division, set up your "box" with these coefficients on the top: -1, 0, 0, 180, 0. Put a 6 outside the box as the divisor.
You can only use synthetic division when you are dividing by something in the form of $x \pm n$. Always put $- \left(n\right)$ outside of the box.

A picture of my work is attached.
I circled the terms I focus on in each step:
Bring the first number down; here, that's $- 1$.

Step 2: Multiply $- 1$ by 6 and put the result under the next coefficient. Then add the column: $0 + - 6 = - 6$.

Step 3: Multiply that answer by 6:
$- 6 \cdot 6 = - 36$ and write it under the next coefficient.
Add those numbers: $0 + - 36 = - 36$.

Step 4:Multiply that result by $6$:
$6 \cdot - 36 = - 216$ and write the result in the fourth column. Add those numbers: $180 + - 216 = - 36$

Step 5: Multiply: $- 36 \cdot 6 = - 216$ and Add: $180 + - 216 = - 36$

Finally, Multiply $- 36 \cdot 6 = - 216$ and Add: $0 + - 216 = - 216$

This last number is the remainder. The remainder should always be divisor of the problem (In this case, x-6).

On the bottom row, you now have the coefficients of the answer: -1, -6, -36, -36, -216.
We know that ${x}^{4} / x = {x}^{3}$. Therefore, the first number is the coefficient of the ${x}^{3}$ term. The next goes with ${x}^{2}$, and so on. The remainder follows the constant, and you get the answer

$- {x}^{3} - 6 {x}^{2} - 36 x - 36 - \frac{216}{x - 6}$.

Finally, synthetic division seems a little magical, so I recommend watching Dr. Khan's videos on synthetic division on KhanAcademy. He works through a problem to show why it works. Good luck!