Objective

Warm-Up

Polynomial Roots & Complex Numbers Review

Review of Polynomial Roots and Complex Numbers

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Objective

SWBAT put together what they have learned about polynomials roots and complex numbers.

Big Idea

Arithmetic with polynomials and complex numbers are performed in similar ways, but powers of i have a meaning distinct from powers of variables.

Lesson Author

Colleen Werner

Ipswich, MA

Grade Level

Eleventh grade

Subjects

maximum points

Math

minimum points

increasing function

polynomial functions

complex conjugates

fundamental theorem of algebra

Standards

HSN-CN.A.1

Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.

HSN-CN.C.7

Solve quadratic equations with real coefficients that have complex solutions.

HSN-CN.C.9

(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

HSA-APR.A.1

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

HSA-APR.B.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

HSA-APR.B.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

HSA-APR.C.4

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

HSA-APR.D.6

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

HSA-CED.A.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

HSA-CED.A.3

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

HSF-IF.C.7c

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

MP1

Make sense of problems and persevere in solving them.

MP2

Reason abstractly and quantitatively.

MP3

Construct viable arguments and critique the reasoning of others.

MP4

Model with mathematics.

MP5

Use appropriate tools strategically.

MP6

Attend to precision.

MP7

Look for and make use of structure.

Warm-Up

15 minutes

For homework, students completed Complex Numbers Intro, which was a substantial assignment. As I check this assignment, I display the solutions on the overhead and ask students to mark on their packets any questions they missed. After I check everyone's work we review the assignment to prepare for the day's review and the next day's summative assessment.

The warm up is an additional set of complex number questions - mostly about simplifying powers of i. I send my students the formative assessment Warm-up- Simplifying Powers of i with the TI NSpire Navigator so that we can all see how the class is progressing in their ability to simplify complex expressions.

Polynomial Roots & Complex Numbers Review

75 minutes

In this lesson, I give students a chance to put their new understanding of imaginary numbers together with what they learned earlier about polynomials. They will be assessed on this material in a quiz the following day.

The activity Waterfall Trivia- Polynomials and Imaginary Roots is a self-advancing powerpoint with three rounds of 5 questions each. In each round, the question is displayed for five minutes. Students copy the problem onto a sheet of paper and work on it independently for five minutes [MP1]. They continue doing this until all 5 problems in the set have been displayed. At the end of each round, students submit their papers to me so I can tally up the points. The top three scorer in all three rounds win a prize, which is candy or some type of school supply.

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Algebra II Â» The Complex Number System