I am trying to implement the QR Code Specification (ISO/IEC 18004:2015) and it contains some polynomial division (to calculate error codes). I'm trying to teach myself polynomial division but am not getting the same answer as the examples in the specification. The specification shows the remainder as being positive.
But when I work it out through long division, my remainder is negative, and in a different form.
My working out is below
Can anyone suggest what I'm doing wrong?
$$ \left( x^{14} + x^{13} + x^{12} \right) = \left( x^{12} + x^{11} + x^{10} + x^{9} + x^{8} + x^{5} + x^{2} + 1 \right) \cdot \left( x^{2} \right) + \left( - x^{11} - x^{10} - x^{7} - x^{4} - x^{2} \right) $$ is correct. For some reason you typed the degree twelve piece in two places.