Rational candidates and exact root finding

Rational candidates and exact root finding

Use the rational zero theorem as a search tool for polynomial equations, then verify candidates and finish the factorization. This precalculus set emphasizes the difference between a possible rational zero and an actual root.

Tasks include complete candidate lists, monic polynomials, rational coefficients, zero constant terms and a cubic whose remaining roots are irrational. A contextual dimension problem adds the requirement that an algebraic solution satisfy the original physical assumptions. Counterexamples show why a polynomial need not have any rational zeros.

Prerequisites are the factor and remainder theorems, polynomial division, factoring and the quadratic formula. Single-select and multiselect questions progress from selecting candidates to combining methods and checking complete solution sets. Use exact arithmetic; no numerical root-finding software is required.

Try a candidate by substitution or synthetic division, record the factor only after a zero remainder, and continue solving the quotient. Worked explanations show which theorem hypotheses matter and why tempting alternatives fail. The set tests exact solvable examples and the limits of a rational search, rather than suggesting that every polynomial can be factored into rational linear factors.