Suppose you have factored it and x comes out to be some x'. Then what you want is a line that passes through (x', f(x')) with slope 1. Clearly, this line is unique. Consider the usual form of the equation of a line: y = mx + c. Here m is the slope and c is the intercept of the line with the y axis. You already have m=1, x=x', y=f(x'). So you can solve for c to get the equation.