Question

How do you use synthetic division to divide `(2x^2 – 23x + 63) ÷ (x – 7)`?

Answer 1

(see below for details of synthetic division method)
`2x^2-23x+63)div (x-7) = (2x-9) R: 0`

Explanation:

Set up for synthetic division by monic binomial divisor:

row [1]: the coefficients of the terms of the dividend polynomial (in descending exponent sequence)

row [3]: the negative of the constant from the divisor (the `xx` sign may be omitted; it is simply there as a reminder for later processing)
and
a copy of the first dividend coefficient

#{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row [1]"), (,"|",,,,color(white)("XXXX")"row [2]"), ("-----",,"-----","-----","-----",), (xx (+7),"|",2,,,color(white)("XXXX")"row [3]") :}#

`"--------------------------------------------------------------------------------"`

Repeat until you have entered a value in the last column of row [3]

Multiply the last value entered in row [3] by the negative of the divisor coefficient and write the product in the next column of row [2].

Add the values in rows [1] and [2] to get the next column value for row [3]
#{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row [1]"), (,"|",,color(white)("X")14,,color(white)("XXXX")"row [2]"), ("-----",,"-----","-----","-----",), (xx (+7),"|",2,-9,,color(white)("XXXX")"row [3]") :}#

`"--------------------------------------------------------------------------------"`

#{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row [1]"), (,"|",,color(white)("X")14,-63,color(white)("XXXX")"row [2]"), ("-----",,"-----","-----","-----",), (xx (+7),"|",color(blue)(2),color(blue)(-9),color(white)("XX")color(red)(0),color(white)("XXXX")"row [3]") :}#

`"--------------------------------------------------------------------------------"`

The `color(red)("last sum")` is the Remainder
The `color(blue)("preceding sums")` are the coefficients of the reduced quotient.

i.e. the final answer is `color(blue)(2)xcolor(blue)(-9)` with a remainder of `color(red)(0)`