Question

How do you divide `(6x^4+9x^3-2x^2+2x-7)/(4x-2)`?

Answer 1

Use either synthetic division or polynomial long division to get
`3/2x^3+3x^2+x+1` with remainder of `(-5)`

Explanation:

synthetic division
#{: (,,6,+9,-2,+2,-7), (+,,,color(white)("X")3,color(white)("X")6,color(white)("X")2,color(white)("X")2), (,,"----","----","---","----","---"), (/ (4),"|",6,12,4,4,color(red)((-5))), (xx (2),"|",color(blue)(3/2),color(blue)(3),color(blue)(1),color(blue)(1),) :}#

Polynomial long division
#{: (,,color(blue)(3/2x^3),color(blue)(+3x^2),color(blue)(+x),color(blue)(+1),), (,,"----","----","----","----","----"), (4x-2,")",6x^4,+9x^3,-2x^2,+2x,-7), (,,6x^4,-3x^3,,,), (,,"----","-----",,,), (,,,12x^3,-2x^2,,), (,,,12x^3,-6x^2,,), (,,,"-----","-----",,), (,,,,4x^2,+2x,), (,,,,4x^2,-2x,), (,,,,"-----","-----",), (,,,,,4x,-7), (,,,,,4x,-2), (,,,,,"----","----"), (,,,,,,color(red)(-5)) :}#