Question

What are common mistakes students make with synthetic division?

Answer 1

Common synthetic division mistakes:
(I have assumed that the divisor is a binomial; since that is by far the most common situation).

Explanation:

Omitting `0` valued coefficients
Given an expression `12x^5-19x^3+100`
It is important to treat this as `12x^5color(red)(+0x^4)-19x^3color(red)(+0x^2)color(red)(+0x)+100`
So the top line looks like:
`color(white)("XXX")12 +0 -19 +0 +0 +100`

Not negating the constant term of the divisor.
For example if the divisor is `(x+3)`
then the multiplier must be `(-3)`

Not dividing by or dividing at the wrong time by the leading coefficient.
If the binomial divisor is not monic, then the sum of the terms must be divided by the leading coefficient before the result is multiplied to give the second term of the next column.

For example `(12^5-19x^3+100) div (2x+3)`
should be set-up as
#{: (,"|",12,+0,-19,+0,+0,+100), (,"|",,,,,,), (,,"-----","-----","-----","-----","-----","-----"), (div2,"|",,,,,,), (xx(-3),"|,,,,,,) :}#