Question

I need to know how to solve this using synthetic division

Answer 1

Recall, first off, you sort the terms in descending order, in this case, they already are in descending order.  Then you use the coefficients.  first off you drop down the first value, and take it from there,

$\bf p(x)=\underline{3} x^3\underline{-4} x^2\underline{-5} x\underline{+1}\qquad a=2\\\\\\ \begin{array}{r|rrrrrrl} 2&&3&-4&-5&1\\ &&&6&4&-2\\ -&&--&--&--&--\\ &&3&2&-1&\boxed{-1}&\impliedby remainder \end{array} \\\\\\ 3x^2+2x-1x+\boxed{\frac{-1}{3x^3-4x^2-5x+1}} \\\\\\ 3x^2+2x-x-\frac{1}{3x^3-4x^2-5x+1}$