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Polynomials practice

Analyzing and solving polynomial equations worksheets

Analyze and solve six expanded monic polynomial equations of selected degrees two through six. Each exact key separates degree and Descartes possibilities from rational candidates and the complete roots with multiplicity.

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For each equation, analyze possible root counts, sign changes and rational candidates, then solve for every exact root with multiplicity.

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Analyze possibilities, then solve the equation

Each expanded equation connects four distinct analyses: degree and conjugate pairing bound real and nonreal counts, Descartes gives possible positive and negative counts, and the Rational Root Theorem supplies candidates. Exact solving then identifies the actual roots and multiplicities.

Keep possibility lists separate from actual answers. Skip zero coefficients when counting sign changes, test rational candidates exactly, and use conjugate pairs to complete irrational and nonreal roots before rebuilding the polynomial.

Learning objective

What this activity builds

Combine degree, conjugate pairing, Descartes’ Rule of Signs and the Rational Root Theorem, then solve for every exact root.

Worked example

Analyze and solve x³ − x² + x − 1 = 0.

Complex total 3; possible real/nonreal counts (3,0) or (1,2); V(P)=3 gives 3 or 1 positive, V(P(−x))=0 gives 0 negative; rational candidates ±1; roots 1, i and −i.

The degree and sign rules give possibilities. Factoring by grouping gives (x − 1)(x² + 1), which identifies the actual roots.

Teaching notes

Make the practice count

  • Count roots with multiplicity; every real root is also a complex root.
  • Skip zero coefficients when counting sign changes in P(x) and P(−x).
  • Rational Root Theorem values are candidates, not guaranteed roots.
  • Use conjugate pairs for real irrational and nonreal roots.
  • Check the final root multiset by rebuilding the monic polynomial.

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Choose any nonempty set of degrees 2 through 6. Every six-question sheet combines root-count possibilities, Descartes sign counts, rational candidates and exact solved roots.

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Math SheetNameScore/ 6TeacherDateAnalyzing and Solving Polynomial EquationsFor each equation, analyze possible root counts, sign changes and rational candidates, then solve for every exact root withmultiplicity.1.x6+x5-11x4+x3-14x2-2x+24=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):2.x4-4x3+3x2+4x-4=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):3.x3+5x2+16x+80=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):4.x6-2x5-26x4+50x3+169x2-288x-144=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):5.x2-2x-15=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):6.x5+6x4+34x3+96x2-16x-208=0Complex / possible real-nonreal:V(P(x)) / possible positive:V(P(-x)) / possible negative:Rational candidates:Exact roots (with multiplicity):math-sheet.comPage 1 / 1

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