# ALEX Classroom Resource

## Precalculus and Advanced Topics Module 3, Topic A: Polynomial Functions and the Fundamental Theorem of Algebra

Classroom Resource Information

Title:

Precalculus and Advanced Topics Module 3, Topic A: Polynomial Functions and the Fundamental Theorem of Algebra

URL:

Content Source:

EngageNY
Type: Lesson/Unit Plan

Overview:

Module 3, Topic A brings students back to the study of complex roots of polynomial functions.  Students briefly review quadratic and cubic functions and then extend familiar polynomial identities to both complex numbers and to general polynomial functions. Students use polynomial identities to find square roots of complex numbers. The binomial theorem and its relationship to Pascal’s Triangle are explored using roots of unity. The topic concludes with students’ use of Cavalieri’s principle to derive formulas for the volume of the sphere and other geometric solids.

Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to the Algebra II and Algebra II with Trigonometry Alabama Courses of Study.

Content Standard(s):
 Mathematics MA2015 (2016) Grade: 9-12 Algebra II 5 ) (+) Extend polynomial identities to the complex numbers. Example: Rewrite x2 + 4 as (x + 2i)(x - 2i). [N-CN8] Mathematics MA2015 (2016) Grade: 9-12 Algebra II 6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9] Mathematics MA2015 (2016) Grade: 9-12 Precalculus 13 ) (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [A-APR5] Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 5 ) (+) Extend polynomial identities to the complex numbers. [N-CN8] Example: Rewrite x2 + 4 as (x + 2i)(x - 2i). Mathematics MA2015 (2016) Grade: 9-12 Algebra II with Trigonometry 6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9] Mathematics MA2015 (2016) Grade: 9-12 Precalculus 37 ) (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. [G-GPE3] Mathematics MA2015 (2016) Grade: 9-12 Precalculus 38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]
Tags: algebra, binomial theorem, Cavalieris principle, complex numbers, complex roots, cubic function, ellipses, foci, Fundamental Theorem of Algebra, hyperbolas, Pascals Triangle, polynomial functions, quadratic function, sphere, square roots, volume
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