Question

How do you use synthetic division to divide `(4x^2 - 2x + 6) (2x - 3) ^ -1`?

Answer 1

`(2x+2)` with remainder `= 12`

Explanation:

`(4x^2-2x+6)*(2x-3)^(-1)`
is equivalent to `(4x^2-2x+6) div (2x-3)`

Using synthetic division (note that there may be slight variations in the structure depending upon how this is being taught):

Dividend: `color(red)(4)x^2color(red)(-2)xcolor(red)(+6)`
Divisor: `color(blue)(2)xcolor(blue)(-3)`

#{: (,"|",color(red)(4),color(red)(-2),color(red)(+6)), (,"|",,6,6), (,,"-----","-----","-----"), (/color(blue)(2),"|",4,4,12), (color(white)("XX")color(cyan)(+3),"|",2,2,) :}#
Note the change in the sign of the constant term of the divisor.

"Bring down" the `4`
Divide by the `/2` to get `2`
Multiply by `+3` to get `6`

#{: (,"|",4,-2,+6), (,"|",,6,), (,,"-----","-----","-----"), (/2,"|",4,,), (color(white)("XX")+3,"|",2,,) :}#

Add the `-2` and `6` to get `4`
Divide by the `/2` to get `2`
Miultiply by `+3` to get `6`

#{: (,"|",4,-2,+6), (,"|",,6,6), (,,"-----","-----","-----"), (/2,"|",4,4,), (color(white)("XX")+3,"|",2,2,) :}#

Add the `+6` and `6` to get the final remainder `12`

#{: (,,x^2,x^1,x^0), (,"|",4,-2,+6), (,"|",,6,6), (,,"-----","-----","-----"), (/2,"|",4,4,color(green)((12))), (color(white)("XX")+3,"|",2,2,), (,,x^1,x^0,) :}#
Note: the additional rows showing corresponding powers of `x` are not really part of the synthetic division (but hopefully add some clarity).