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

1 Answer
Sep 2, 2017

(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. enter image source here
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!