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

Concepts

Lessons

Practice

Videos

Handouts

19. What is a polynomial?

20. Put a polynomial into standard form, identify the degree and leading coefficient

21. Factor polynomial equations completely using one or more of the following methods: factor out a greatest common factor, use the diamond/box method, split the middle, factor the difference of two squares, use substitution to make a complex problem easier.

22. Determine the multiplicity of factors and how it affects the shape of the graph.

23. Understand how the leading coefficient affects end behavior

24. Understand how the degree determines start/end behavior, the number of turns, the number of roots and the number of x-intercepts.

25. Sketch polynomial graphs from factored form without a calculator.

26. Find a polynomial equation in factored form from its graph including the “a”.

quiz

27. Use the quadratic formula to find real and complex roots of polynomials

28. Understand that i =√-1 . Use this to simplify square roots.

29. Use the discriminant to describe the nature of the roots.

30. Use the sum and product of the roots to find a quadratic equation in standard form with integer coefficients

31. Simplify powers of i and perform operations on complex numbers.

32. Solve equations with complex solutions:

28. Use the degree to determine the number of real and complex roots.

Quiz

29. Polynomial division: grid method, synthetic division

30. Using polynomial division to factor.

31. Finding all roots given only an equation and a graph.

32. Factor the sum and difference of two cubes.

33. Synthetic Division (you can use the box method instead)

Factoring Higher Power Polynomials
Factoring by Grouping
Fundamental Theorem of Algebra (and Polynomials)
Polynomial Equations
Polynomial Graphs (zeros, multiplicity, end behavior, graph behavior near roots)

Polynomial Remainder Theorem (and Factor Theorem)

Terms and Degree
Finding Zeros on the Calculator
Division of Polynomials
The Remainder and Factor Theorems
Factoring Polynomials Using the Factor Theorem
Methods of Factoring Polynomials
Solving Polynomial Equations
Solving Polynomial Inequalities
Subtracting Polynomials
Multiplying Polynomials
Dividing Polynomials
The Remainder Theorem
Introduction to Polynomial Functions
Characteristics of Polynomial Functions
Graphs of Polynomial Functions in Factored Form
Equations of Polynomial Functions in Factored Form

Factoring Higher Power Polynomials
Factoring and Perfect Cubes
Factoring by Grouping
Polynomial Long Division
Synthetic Division
Polynomial Remainder Theorem
Fundamental Theorem of Algebra
Polynomial Equations
Polynomial Identities
Polynomial Graphs
(zeros, multiplicity, end behavior, graph
behavior near roots)
Practice with Factoring by Grouping
Practice with Polynomial Division
Practice with Polynomial Equations
(multiple choice)
Practice with More Polynomial Equations
(solutions shown)
Practice with Polynomial Identities
Practice Graphing Polynomial Functions
Practice Working with Polynomials
Polynomials (definition, degree, leading term, constant, standard form, closure)
Polynomials - Divide (only monomial or factorable for binomial)
To refresh basic skills with Factoring
Polynomials, see Factoring section from
Algebra 1.
General Features of Function Graphs
(intercepts, positive/negative,
increase/decrease)
More Features of Function Graphs (continuous, maximum/minimum, symmetry, end behavior)
Transformations of Functions
For factoring practice go here
Factoring:
http://www.phschool.com/webcodes10/index.cfm?wcprefix=aga&wcsuffix=0602&area=view
Complex Numbers:
http://www.phschool.com/webcodes10/index.cfm?wcprefix=aga&wcsuffix=0506&area=view
Discriminant:
http://www.phschool.com/webcodes10/index.cfm?wcprefix=ag
Roots:
(SOAP)
(You can use box method here)
Square roots in terms of i
Powers of i
Complex conjugate
Nature of roots of a quadratic
Sum and product of roots of a quadratic
Quadratic equation given the sum and product of its roots
Completing the square
Factor polynomials completely
Even and odd functions

Even and Odd Polynomial Functions
Practice with Factoring by Grouping
Practice with Polynomial Division
Practice with Polynomial Equations (multiple choice)
Practice with More Polynomial Equations (solutions shown)
Practice Graphing Polynomial Functions
Practice Working with Polynomials
Polynomials (definition, degree, leading term, constant, standard form, closure)

How Do You Determine the Zeros of a Polynomial Function from a Table of Values?

##### The Fundamental Theorem of Algebra

How Do You Solve a Word Problem Using the Quadratic Formula?

How Do You Multiply Pure Imaginary Numbers?

How Do You Multiply Complex Numbers Using FOIL?

How Do You Divide Complex Numbers Using Conjugates?

How Do You Find Values for x and y to Make Two Complex Numbers Equal?

How Do You Subtract Complex Numbers?

How Do You Add Complex Numbers?

How Do You Find Higher Powers of i?

Using a polynomial function and its factors solve problems VID

Finding the zeros of a polynomial function

Writing a polynomial from its zeros

Solving polynomial equations by graphing VID

Factoring a sum or difference of cubes

Solving polynomial equations of higher degrees using factoring and the quadratic formula

Solving polynomial equations of higher degrees using factoring

Using the Rational Root Theorem to find roots VID

Solving equations using the Rational Root Theorem

Using the Irrational Root Theorem to find irrational roots

Writing a polynomial from its roots

Using the Fundamental Theorem of Algebra VID

Solving quadratic equations using square roots VID

Solving quadratic equations by factoring VID

Solving quadratic equations using completing the square VID

Solving quadratic equations using completing the square when a ≠ 1

Solving real-world problems using the quadratic formula

Choosing the appropriate method to solve quadratic equations

Using the discriminant to find the number of solutions and solve problems VID

Finding complex solutions VID

Finding Imaginary Root Theorem VID

-Polynomial Packet Part I.pdf

-Polynomial Packet Part 2.pdf

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