Question

How do you use synthetic division to divide `(x^2 + 13x + 40) ÷ (x + 8)`?

Answer 1

`(x^2+13x+40)div(x+8) = (x+5)` with a remainder of `0`
(see below for synthetic division method for this example)

Explanation:

The top row are the coefficients of the dividend expression
The value on the left one row down is the negative of the constant in the divisor (i.e. the value of `x` that would make `(x+8)=0`).
The value on the very bottom (which will become our result row) left is always `0` (the result we have determined so far).
#{: (," | ",1,13,40), (-8," | ", 0,,), ("-----","-+-","-----","-----","-----"), (0,,,,) :}#

Add the two numbers in the completed column above the line to get a sum on the bottom row (in this case `1`):
Multiply `(-8)` and the next number in the result row (in this case `1`) and write the result under the next dividend coefficient.
#{: (," | ",1,13,40), (-8," | ", 0,-8,), ("-----","-+-","-----","-----","-----"), (0,,1,,) :}#

Add the two numbers in the completed column above the line to get a sum on the bottom row (in this case `5`):
Multiply `(-8)` and the next number in the result row (in this case `5`) and write the result under the next dividend coefficient (in this case `(-40)`.
#{: (," | ",1,13,40), (-8," | ", 0,-8,-40), ("-----","-+-","-----","-----","-----"), (0,,1,5,) :}#

Addition of the final column gives the remainder (you might want to write it on a separate line.
Here is the final result with labels on entries for clarity.
#{: (,,x^2,x^1,x^0), (," | ",1,13,40), (-8," | ", 0,-8,-40), ("-----","-+-","-----","-----","-----"), (0,,1,5,), (x^2,,x^1,x^0,), (,,,,R=0) :}#