Question

How do you divide `(-x^4-x^3-x^2-7x-7)/(x-2)`?

Answer 1

Either use synthetic division or polynomial long division to get:
`color(white)("XXX")-x^3-3x^2-7x-21` with a remainder of `(-49)`

Explanation:

I will so the solution using synthetic division (since it's faster, takes less space, and often less error prone)

#{: (color(brown)("[1]"),,"|",color(brown)(x^4),color(brown)(x^3),color(white)("X")color(brown)(x^2),color(white)("XX")color(brown)(x^1),color(white)("XX")color(brown)(x^0)), (color(brown)("[2]"),,"|",-1,-1,-1,color(white)("X")-7,color(white)("X")-7), (color(brown)("[3]"),,"|",,-2,-6,-14,-42), (,bar(color(white)("XXX")),bar(color(white)("XXX")),bar(color(white)("XXX")),bar(color(white)("XXX")),bar(color(white)("XXX")),bar(color(white)("XXX")),bar(color(white)("XXX"))), (color(brown)([4]),xx (color(red)(+)2),"|",color(blue)(-1),color(blue)(-3),color(blue)(-7),color(blue)(-21), color(green)(-49)), (color(brown)([5]),,,color(white)("X")color(brown)(x^3),color(white)("X")color(brown)(x^2),color(white)("X")color(brown)(x^1),color(white)("X")color(brown)(x^0),color(white)("X")color(brown)("R")) :}#

Notes:
The entries in `color(brown)("brown")` would not normally be written down; they are only here for clarity.

Remember if dividing by `(xcolor(red)(-)2)`, the synthetic division multiplier must be `(color(red)(+)2)`