Polynomials practice
Connect factored forms, complete exact zero lists, primitive integer polynomials and complete rational factorizations in twelve balanced Algebra 2 problems.
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Find exact zeros from factors, construct least-degree integer polynomials from zeros, and factor expanded polynomials completely before solving.
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Customize this worksheetThe first four problems start from complete factors, the next four build a primitive integer polynomial from exact zeros, and the final four start from an expanded polynomial that must be factored completely.
Use conjugate closure for irrational and nonreal zeros, preserve every multiplicity, and clear denominators only after forming the monic product. The worked key checks the primitive integer result exactly.
Learning objective
Move exactly among factors, zeros with multiplicity and primitive positive-leading integer polynomials.
Worked example
8x³ - 40x² + x - 5.
The conjugate pair gives x² + 1/8. Multiply by x - 5, clear denominators, divide any common coefficient factor and keep the leading coefficient positive.
Teaching notes
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Each English-only sheet has 12 tasks in three fixed sections. Imaginary mode uses all-nonreal zeros and even degrees.
A useful next step
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Use degree and conjugate pairing only—not polynomial solving—to list possible real and nonreal complex-root counts, counting multiplicity and including real roots in the complex total. Complete rational-coefficient root lists and construct least-degree monic integer polynomials.
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