How do you use synthetic division to divide 180x−x4x−6?

1 Answer
LizZ
Sep 2, 2017

180x−x4x−6=−x3−6x2−36x−36−216x−6

Explanation:

Write 180x−x4 in standard form, with the x4 term first and 0 as the coefficient for the "missing" terms: −x4+0x3+0x2+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 x4x=x3. Therefore, the first number is the coefficient of the x3 term. The next goes with x2, and so on. The remainder follows the constant, and you get the answer

−x3−6x2−36x−36−216x−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!

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