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Module 1: Polynomial, rational, and radical relationships

Topic A: Polynomials—from base ten to base X

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Lessons 1-2: No content yet
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Intro to long division of polynomials
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Dividing polynomials: long division
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Divide polynomials by x (with remainders)
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Divide polynomials by monomials (with remainders)
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Dividing polynomials with remainders
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Lessons 7-11: No content yet
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Practice
Divide polynomials by monomials (with remainders)Get 3 of 4 questions to level up!
Divide polynomials by linear expressionsGet 3 of 4 questions to level up!
Divide polynomials with remaindersGet 3 of 4 questions to level up!

Topic B: Lessons 12-13: Factorization

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Factoring quadratics: common factor + grouping
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Factoring quadratics: negative common factor + grouping
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Factoring two-variable quadratics
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Factoring two-variable quadratics: rearranging
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Factoring two-variable quadratics: grouping
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Factoring using the perfect square pattern
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Factoring using the difference of squares pattern
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Factoring difference of squares: two variables (example 2)
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Factoring higher-degree polynomials
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Factoring sum of cubes
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Factoring difference of cubes
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Finding zeros of polynomials (1 of 2)
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Finding zeros of polynomials (2 of 2)
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Finding zeros of polynomials (example 2)
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Practice
Factor polynomials: quadratic methodsGet 3 of 4 questions to level up!
Factor polynomials: quadratic methods (challenge)Get 3 of 4 questions to level up!
Factor polynomials using structureGet 3 of 4 questions to level up!
Zeros of polynomials (with factoring)Get 3 of 4 questions to level up!

Topic B: Lesson 14: Graphing factored polynomials

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Zeros of polynomials & their graphs
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Zeros of polynomials & their graphs
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Positive & negative intervals of polynomials
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Intro to end behavior of polynomials
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End behavior of polynomials
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End behavior of functions & their graphs
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Graphs of polynomials
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Graphs of polynomials: Challenge problems
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Practice
Zeros of polynomials (factored form)Get 3 of 4 questions to level up!
Positive & negative intervals of polynomialsGet 3 of 4 questions to level up!
End behavior of polynomialsGet 3 of 4 questions to level up!

Topic B: Lesson 15: Structure in graphs of polynomial functions

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Function symmetry introduction
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Function symmetry introduction
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Even and odd functions: Graphs
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Even/odd functions & numbers
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Even & odd polynomials
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Symmetry of polynomials
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Practice
Even and odd functions: Graphs and tablesGet 3 of 4 questions to level up!
Even & odd functions: EquationsGet 3 of 4 questions to level up!

Topic B: Lessons 16-21: Remainder theorem and modeling

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Lessons 16-17: No content yet
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Intro to the Polynomial Remainder Theorem
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Remainder theorem: finding remainder from equation
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Remainder theorem: checking factors
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Remainder theorem: finding coefficients
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Proof of the Polynomial Remainder Theorem
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Lessons 20-21: No content yet
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Practice
Remainder theoremGet 3 of 4 questions to level up!

Topic C: Lessons 22-23: Simplifying rational expressions

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Intro to rational expression simplification
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Intro to rational expressions
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Intro to simplifying rational expressions
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Simplifying rational expressions: common monomial factors
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Simplifying rational expressions: common binomial factors
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Simplifying rational expressions: opposite common binomial factors
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Simplifying rational expressions (advanced)
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Simplifying rational expressions: grouping
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Simplifying rational expressions: higher degree terms
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Simplifying rational expressions: two variables
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Lesson 23: No content yet
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Practice
Simplify rational expressions: common monomial factorsGet 3 of 4 questions to level up!
Simplify rational expressions: common binomial factorsGet 3 of 4 questions to level up!
Simplify rational expressions (advanced)Get 3 of 4 questions to level up!

Topic C: Lesson 24: Multiplying and dividing rational expressions

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Multiplying & dividing rational expressions: monomials
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Multiplying rational expressions
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Dividing rational expressions
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Multiplying rational expressions
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Dividing rational expressions
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Multiplying rational expressions: multiple variables
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Dividing rational expressions: unknown expression
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Practice
Multiply & divide rational expressions (basic)Get 3 of 4 questions to level up!
Multiply & divide rational expressionsGet 3 of 4 questions to level up!
Multiply & divide rational expressions (advanced)Get 3 of 4 questions to level up!

Topic C: Lesson 25: Adding and subtracting rational expressions

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Adding & subtracting rational expressions: like denominators
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Intro to adding & subtracting rational expressions
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Intro to adding rational expressions with unlike denominators
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Adding rational expression: unlike denominators
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Subtracting rational expressions: unlike denominators
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Least common multiple
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Least common multiple: repeating factors
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Subtracting rational expressions: factored denominators
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Least common multiple of polynomials
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Adding & subtracting rational expressions (advanced)
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Subtracting rational expressions
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Nested fractions
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Practice
Add & subtract rational expressions: like denominatorsGet 3 of 4 questions to level up!
Add & subtract rational expressions: unlike denominatorsGet 3 of 4 questions to level up!
Least common multipleGet 3 of 4 questions to level up!
Add & subtract rational expressions: factored denominatorsGet 3 of 4 questions to level up!
Add & subtract rational expressionsGet 3 of 4 questions to level up!
Nested fractionsGet 3 of 4 questions to level up!

Topic C: Lessons 26-27: Rational equations

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Rational equations intro
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Equations with one rational expression (advanced)
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Equations with rational expressions
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Equations with rational expressions (example 2)
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Rational equations word problem: combined rates
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Rational equations word problem: combined rates (example 2)
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Rational equations word problem: eliminating solutions
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Practice
Rational equations introGet 3 of 4 questions to level up!
Rational equations (advanced)Get 3 of 4 questions to level up!
Rational equationsGet 3 of 4 questions to level up!

Topic C: Lessons 28-29: Radical equations

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Intro to square-root equations & extraneous solutions
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Intro to solving square-root equations
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Solving square-root equations
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Solving square-root equations: one solution
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Solving square-root equations: two solutions
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Solving square-root equations: no solution
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Square-root equations intro
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Equation that has a specific extraneous solution
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Extraneous solutions of radical equations
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Solving cube-root equations
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Practice
Square-root equations introGet 3 of 4 questions to level up!
Square-root equationsGet 3 of 4 questions to level up!
Extraneous solutions of equationsGet 3 of 4 questions to level up!

Topic C: Lesson 30: Linear systems in three variables

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Intro to linear systems with 3 variables
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Solving linear systems with 3 variables
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Solving linear systems with 3 variables: no solution
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3-variable linear system word problem
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Topic C: Lesson 31: Systems of equations

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Quadratic systems: graphical solution
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Quadratic system with no solutions
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Quadratic systems: both variables are squared
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Quadratic systems: a line and a circle
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Topic C: Lesson 32: Graphing systems of equations

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Interpreting equations graphically
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Interpreting equations graphically (example 2)
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Solving equations graphically (1 of 2)
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Solving equations graphically (2 of 2)
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Solving equations graphically
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Practice
Interpret equations graphically Get 3 of 4 questions to level up!
Solve equations graphicallyGet 3 of 4 questions to level up!

Topic C: Lessons 33-35: Parabolas

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Intro to focus & directrix
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Equation of a parabola from focus & directrix
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Focus & directrix of a parabola from equation
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Parabola focus & directrix review
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Practice
Equation of a parabola from focus & directrixGet 3 of 4 questions to level up!

Topic D: Lesson 37: Complex numbers intro

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Intro to the imaginary numbers
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Intro to the imaginary numbers
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Simplifying roots of negative numbers
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i as the principal root of -1
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Intro to complex numbers
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Plotting numbers on the complex plane
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The complex plane
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Practice
Simplify roots of negative numbersGet 3 of 4 questions to level up!
Parts of complex numbersGet 3 of 4 questions to level up!
Classify complex numbersGet 3 of 4 questions to level up!
Plot numbers on the complex planeGet 3 of 4 questions to level up!

Topic D: Lesson 37: Complex number operations

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Adding complex numbers
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Subtracting complex numbers
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Multiplying complex numbers
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Multiplying complex numbers
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Powers of the imaginary unit
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Powers of the imaginary unit
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Practice
Add & subtract complex numbersGet 3 of 4 questions to level up!
Graphically add & subtract complex numbers Get 3 of 4 questions to level up!
Multiply complex numbers (basic)Get 3 of 4 questions to level up!
Multiply complex numbersGet 3 of 4 questions to level up!
Powers of the imaginary unitGet 3 of 4 questions to level up!

Topic D: Lessons 38-40: Complex roots and the fundamental theorem of algebra

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Solving quadratic equations: complex roots
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Complex numbers & sum of squares factorization
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Factoring sum of squares
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The Fundamental theorem of Algebra
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Quadratics & the Fundamental Theorem of Algebra
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Number of possible real roots of a polynomial
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Practice
Solve quadratic equations: complex solutionsGet 3 of 4 questions to level up!
Factor polynomials: complex numbersGet 3 of 4 questions to level up!

About this unit

"Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations." Eureka Math/EngageNY (c) 2015 GreatMinds.org