Polynomials practice

Rational Root Theorem worksheets

List every possible rational zero from the printed leading and constant coefficients, then find exact rational zeros and their multiplicities in twelve degree-two through degree-four problems.

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List all possible rational zeros. For the final six problems, also find every actual rational zero and its multiplicity.

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Build every ±p/q candidate, then test it

Use every positive divisor p of the printed nonzero constant and every positive divisor q of the printed leading coefficient. Reduce all signed fractions and remove duplicates only after reduction.

A possible rational zero is only a candidate. Test candidates exactly; when one is a zero, divide repeatedly by its linear factor to record its multiplicity. Native exercises have degree two through four, nonzero constants and split completely over the rationals.

Learning objective

What this activity builds

Generate the complete reduced ±p/q candidate set and test it exactly to identify every rational zero with multiplicity.

Worked example

List the possible rational zeros of 2x² − 5x + 2, then find the actual zeros.

Possible: ±1, ±2, ±1/2. Actual: 1/2 and 2, each with multiplicity 1.

Use factors of the printed constant over factors of the printed leading coefficient, reduce duplicates, then test each candidate exactly.

Teaching notes

Make the practice count

Use the printed coefficients before extracting any numerical content.
Reduce every p/q and remove duplicates without dropping either sign.
A possible rational zero is only a candidate until substitution or exact division gives remainder zero.
Continue dividing by the same linear factor to determine multiplicity.

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Choose degrees 2, 3 and 4 in any combination. Keys list every reduced signed p/q candidate; the final six also identify each actual rational zero and its multiplicity.

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Remainder theorem worksheets

Use the Remainder Theorem in twelve structured problems: evaluate four polynomials at a point, make four factor decisions from zero remainders, and divide four polynomials by signed monic linear divisors with exact quotient-and-remainder answers.

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