The document discusses the Fundamental Theorem of Algebra and properties of polynomial functions. It states that any polynomial function of degree n will have at least one root in the complex number system. Additionally, a polynomial of degree n will have exactly n roots counting multiplicities. The document defines roots or zeros as solutions to the equation f(x)=0 and provides examples of finding all zeros of a given polynomial function and writing a polynomial function with given complex zeros.

1. Algebra 2
6.7 Using the Fundamental Theorem of Algebra
If f(x) is a _____________________________ of degree n > 0, then the equation f(x) = 0
has at least one __________________ in the set of complex numbers.
• If you count all ______________, _________________________, and
__________________________ solutions, an nth-degree polynomial has
_____________________ solutions.
• Solutions are also called ___________________ or _________________.
EX1: Find all the zeros of f (x) = ξ5 − 2 ξ4 + 8 ξ2 − 13ξ + 6 .
• All ______________ zeros of a function are ____________________________.
• __________________________ zeros only touch the x-axis.
• Non-repeated zeros ______________________ the x-axis.
• Remember, complex roots occur in _________________________.
Using Zeros
EX2: Write a polynomial function of least degree that has a leading coefficient of 1,
and 2 and 1 + I as zeros.
HW p. 369 (26 – 46 even)