The fundamental theorem of algebra

The  exercise appears under the Algebra II Math Mission. This exercise introduces and practices the Fundamental Theorem of Algebra.

Types of Problems
There are three types of problems in this exercise:


 * 1) Based on degree, give number of roots: This problem says that a polynomial has a given degree. The student is asked to determine the number of roots, or real roots, of the polynomial.Ftoalg1.png
 * 2) Given the roots, determine the possible degree: This problem says that a certain polynomial has a certain number of roots. The student is asked to determine which of the values could be the degree from a list of multiple select options.Ftoalg2.png
 * 3) Determine which statements are true about the polynomial: This problem provides a function in some form (for example, graph). The student is asked to analyze the function and determine all statements that are true and indicate them from the multiple select list.Ftoalg3.png

Strategies
Knowledge and comfort with factoring techniques and classification of roots (including complex conjugates) are encouraged to ensure success on this exercise.
 * 1) The fundamental theorem of algebra technically states than any polynomial of degree greater than zero has at least one root.
 * 2) The theorem is more commonly used by one of it's characterizations, namely that a polynomial of degree n has n roots including multiplicities and complex solutions.
 * 3) Multiplicities are counting a repeated root as more than one root.
 * 4) Solutions to an equation are only counted once, although roots are counted more than once if they have multiplicity.

Real-life Applications

 * 1) The fundamental theorem of algebra explains how all polynomials can be broken down, so it provides structure for abstraction into fields like Modern Algebra.