Question

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

Answer 1

`(180x-x^4)/(x-6)=-x^3-6x^2-36x-36-216/(x-6)`

Explanation:

Write `180x-x^4` in standard form, with the `x^4` term first and 0 as the coefficient for the "missing" terms: `-x^4+0x^3+0x^2+180x+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+-n`. Always put `-(n)` 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*6=-36` and write it under the next coefficient.
Add those numbers: `0+ -36=-36`.

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

Step 5: Multiply: `-36*6=-216` and Add: `180+ -216=-36`

Finally, Multiply `-36 * 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^3-6x^2-36x-36-216/(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!