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

1 Answer
Sep 2, 2017

Answer:

#(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!