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Irrational and imaginary root theorem worksheets

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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For root counts, use degree and conjugate pairing only; do not solve. Count multiplicity; complex roots include real roots. Then complete root lists and construct monic integer polynomials.

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Pair every forced conjugate and preserve multiplicity

Degree counts every complex root with multiplicity, including real roots. Nonreal counts must be even, but these possibilities do not identify the displayed polynomial’s actual roots.

For rational coefficients, a + b√d forces a − b√d and a + b√d i forces a − b√d i with matching multiplicity. Construction uses the minimal closure and returns the monic integer polynomial.

Learning objective

What this activity builds

Apply real-irrational and nonreal conjugate closure with multiplicity, then build the unique least-degree monic integer polynomial.

Worked example

A rational-coefficient polynomial has root 1 − √3 with multiplicity 2. What root is forced, and what factor, including multiplicity, is required?

1 + √3 also has multiplicity 2; the factor is (x² − 2x − 2)².

Real irrational roots occur in conjugate pairs. Multiplicity carries to the conjugate, and the pair multiplies to x² − 2ax + a² − b²d.

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  • A degree-n polynomial has n complex roots counting multiplicity; nonreal roots occur in pairs.
  • Possible real/nonreal counts use degree and pairing only and do not claim the displayed polynomial’s actual roots.
  • For rational coefficients, both real irrational and nonreal roots require their conjugates.
  • Do not add a conjugate that is already supplied, and preserve its stated multiplicity.
  • For construction, use one linear factor per integer root and one integer quadratic per conjugate pair.

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Choose any nonempty set of degrees 2 through 6. Each sheet has four possible-count questions, four conjugate-completion questions, and four least-degree constructions.

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Math SheetNameScore/ 12TeacherDateIrrational and Imaginary Root TheoremsFor root counts, use degree and conjugate pairing only; do not solve. Count multiplicity; complex roots include real roots. Thencomplete root lists and construct monic integer polynomials.A. Count possible real and nonreal complex roots1.x2+4x-8Total complex roots:Possible real/nonreal pairs:2.x6+2x5+9x2+3x+8Total complex roots:Possible real/nonreal pairs:3.x4-6x3+7x2-3x-1Total complex roots:Possible real/nonreal pairs:4.x3+8x2+4x-1Total complex roots:Possible real/nonreal pairs:B. Complete each rational-coefficient root list5.Given roots:-1-3i,1-3,4Missing roots:6.Given roots:3-5i,3+5i,-3+3,-2+iMissing roots:7.Given roots:-3+2i,4(multiplicity 2),-4Missing roots:8.Given roots:-4+5,-4Missing roots:C. Construct the least-degree monic integer polynomial9.Given roots:-3+3Least-degree monic polynomial:10.Given roots:2+2i,2+2Least-degree monic polynomial:11.Given roots:3-4i,-4(multiplicity 2),-2Least-degree monic polynomial:12.Given roots:4-2i,-4Least-degree monic polynomial:math-sheet.comPage 1 / 1

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