Unit 03 · 3 Weeks · 7 Lessons
Polynomial Equations, Division & Graphs
How can division, factor relationships, multiplicity, and graph features expose every important part of a polynomial model?
Complete instructional sequence
Detailed Lessons
Lessons include objectives, formulas, concept development, worked examples, guided and independent practice, differentiated pathways, application, misconception analysis, mastery evidence, reflection, and local completion.
Long Division of Polynomials
Polynomial long division mirrors whole-number division: order terms, divide leading terms, multiply, subtract, and repeat. Missing powers should be represented with zero coefficients.
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Lesson 02Synthetic Division & Efficient Evaluation
Synthetic division compresses polynomial long division when the divisor is linear. The final row gives quotient coefficients followed by the remainder.
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Lesson 03Remainder and Factor Theorems
The Remainder Theorem connects evaluation and division. The Factor Theorem states that x−c is a factor exactly when P(c)=0.
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Lesson 04Rational Root Theorem & Candidate Zeros
The Rational Root Theorem produces candidates, not guaranteed zeros. Test candidates efficiently and reduce the polynomial after finding a zero.
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Lesson 05Solving Higher-Degree Polynomial Equations
Higher-degree equations may require grouping, substitution, known factors, or numerical approximation. A complete solution set includes real and complex roots with multiplicity.
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Lesson 06Zeros, Multiplicity & Graph Behavior
Odd multiplicity usually crosses the x-axis; even multiplicity touches and turns. Larger multiplicity flattens the graph near the zero.
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Lesson 07Polynomial Regression, Interpolation & Model Limits
Polynomial models can interpolate or approximate curved data, but higher degree may overfit. Valid modelling includes domain limits, residual review, units, and sensitivity.
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Practice and evidence
Unit Resources
Foundation Practice
Prerequisite support, smaller steps, representation aids, and correction prompts.
Core Practice
Complete standards-level definitive-answer practice and explanation.
Extended Practice
Generalization, proof, parameter analysis, technology, and deeper modelling.
Unit Mastery Check
Fresh randomized questions with scoring and explanations.
Polynomial Graph Reconstruction & Model Defense
Applied investigation with evidence, validation, limitations, and defense.
Family Guide
Plain-language overview, support questions, and evidence expectations.
Teacher Guide
Pacing, differentiation, assessment, misconceptions, and records.
Vocabulary Data
Core English terms with Spanish, French, Arabic, and Portuguese support.
Standards Map
Unit concepts and broad national/international alignment categories.