The Fundamental Theorem of Algebra expresses that any polynomial of degree n will have n roots. Moreover, Descartes’ rule of signs states that the number of real positive and negative roots can be determined through the number of sign changes present within a given polynomial. In order to demonstrate my understanding of the Fundamental Theorem of Algebra and Descartes’ rule of signs, I will provide two polynomials and predict the number of complex roots for each.

Polynomial 1: f(x)=x^4-6x^2+x^3+3x-4

Based on my understanding, the degree let’s me know how many zeros the function has—4 in this case, which I have highlighted in blue. With this in mind, I can now proceed to calculate the number of possible positive and negative real roots.