The Fundamental Theorem of Algebra It turns out that linear factors (=polynomials of degree 1) and irreducible quadratic polynomials are the "atoms", the building blocks, of all polynomials:
Every polynomial can be factored (over the real numbers) into a product of linear factors and irreducible quadratic factors.
The Fundamental Theorem of Algebra was first proved by Carl Friedrich Gauss (1777-1855).
What does The Fundamental Theorem of Algebra tell us? It tells us, when we have factored a polynomial completely:
On the one hand, a polynomial has been completely factored (over the real numbers) only if all of its factors are linear or irreducible quadratic.
On the other hand, whenever a polynomial has been factored into only linear and irreducible quadratics, then it has been factored completely, since both linear factors and irreducible quadratics cannot be factored any further over the real numbers.
What does The Fundamental Theorem of Algebra not tell us? It is not constructive, that is, it does not tell us how to factor a polynomial completely!
In fact, it is not known in general how to factor a polynomial; only techniques for special kinds of polynomials are known. It is even worse than that: The mathematician Evariste Galois (1811-1832) has proved that there will never be a general formula to solve polynomials of degree 5 and higher.
Understand patterns, relations, and functions
High school students' algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. In helping high school students learn about the characteristics of particular classes of functions, teachers may find it helpful to compare and contrast situations that are modeled by functions from various classes. For example, the functions that model the essential features of the situations in figure 7.4 are quite different from one another. Students should be able to express them using tables, graphs, and symbols.

In high school, students should have opportunities to build on these earlier experiences, both deepening their understanding of relations and functions and expanding their repertoire of familiar functions. Students should use technological tools to represent and study the behavior of polynomial, exponential, rational, and periodic functions, among others. They will learn to combine functions, express them in equivalent forms, compose them, and find inverses where possible. As they do so, they will come to understand the concept of a class of functions and learn to recognize the characteristics of various classes.
High school algebra also should provide students with insights into mathematical abstraction and structure. In grades 9–12, students should develop an understanding of the algebraic properties that govern the manipulation of symbols in expressions, equations, and inequalities. They should become fluent in performing such manipulations by appropriate means—mentally, by hand, or by machine—to solve equations and inequalities, to generate equivalent forms of expressions or functions, or to prove general results.
The expanded class of functions available to high school students for mathematical modeling should provide them with a versatile and powerful means for analyzing and describing their world. With utilities for symbol manipulation, graphing, and curve fitting and with programmable software and spreadsheets to represent iterative processes, students can model and analyze a wide range of phenomena. These mathematical tools can help students develop a deeper understanding of real-world phenomena. At the same time, working in real-world contexts may help students make sense of the underlying mathematical concepts and may foster an appreciation of those concepts.