Polynomial & Rational Expressions at the 700-800 Level

Every hard rational item opens with the same unwritten instruction: factor first. Almost nothing at this level survives factoring intact, and once it is factored the item usually answers itself.

Four archetypes account for the bank. Holes vs. vertical asymptotes: a cancelling factor makes a hole, a surviving factor makes an asymptote, and the question asks for a combination — the sum of the hole's coordinates, or the hole's x-value plus the asymptote's — precisely so that each individual feature can be a wrong answer. Extraneous solutions: clearing denominators manufactures roots the domain forbids, so the excluded values get written down before you solve, and "no solution" is a live option. Context models: average cost $(F + vx)/x$, work-and-rate equations where rates add, round trips where average speed is total distance over total time (never the mean of the two speeds), and mixture models where x, the final solute, and the final volume are all choices. Manipulation under time pressure: combining into $(ax+b)/(x^2-1)$, complex fractions where $3 - x = -(x - 3)$ is the entire item, negative exponents, and the factor theorem.

The distractor species are the other coordinate, the extraneous root, the intermediate root, the illegitimate average, and the dropped sign.