What is a polynomial function in algebra?
A polynomial function is a precisely defined expression that is constructed from indeterminates (also called variables) and coefficients. This expression can be evaluated with addition, subtraction, multiplication, and non-negative integer exponents of the variables. Often, polynomial functions are used to encode information about other objects such as matrices and linear operators.
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What Is a Polynomial Function in Algebra?
A polynomial is an algebraic expression that consists of indeterminates and a constant. The coefficients of a polynomial are real numbers or complex numbers. A polynomial is a specific kind of function that can be evaluated with addition, subtraction, multiple-step multiplication, and positive integer exponents.
Graphs of Polynomial Functions:
The simplest way to think about a polynomial function is to graph the function on a coordinate plane, which is a grid with rectangular or square cells. This type of graph is called a function graph because it represents the rule given by a polynomial in one variable.
When you plot a polynomial function, there are four main things that you should notice: turning points, intercepts, end behavior and the Intermediate Value Theorem. These features of a polynomial function are important to note because they determine how a graph will look and how the function will behave as it is solved.
Turning Points:
A turning point on a polynomial function is a point where the graph crosses the x-axis. This happens because the sign of f(x) changes from one side to the other and a maximum or minimum is created on the graph. The number of turning points on a graph can be determined by finding the zeros and determining which ones cross the x-axis.
Intercepts:
An intercept of a polynomial function is essentially a point where the graph crosses the y-axis. This point is usually a convenient place to find when you are drawing a graph. In some cases, you may be able to find an intercept without having to draw the graph, but in most cases it will be easier to do with the help of the graph.
End Behavior of a Function:
The degree and leading coefficient of a polynomial function determine the end behavior of the graph. In general, functions of a certain degree will have graphs with smooth curves and no breaks. For example, a quadratic function with a degree of 2 has a graph that does not have any sharp corners and displays a continuous curve.
In addition, a polynomial function will have a single root at x and a double root at x-+. This is a useful feature of the graph because it shows that the function can be found in both negative and positive values.
As you can see, a polynomial function has many different characteristics that make it unique from other types of graphs. These characteristics are the reason that students need to understand these features in order to be able to interpret the graphs they learn to solve for a polynomial function.
In conclusion, a polynomial function in algebra is an algebraic expression consisting of indeterminates (variables) and coefficients, which can be evaluated using addition, subtraction, multiplication, and non-negative integer exponents. The coefficients are real or complex numbers, and polynomial functions encode information about various mathematical objects.
Graphing a polynomial function provides valuable insights into its behavior. Important features to observe on the graph include turning points, intercepts, end behavior, and the Intermediate Value Theorem. Turning points occur where the graph crosses the x-axis, indicating a change in sign and the creation of a maximum or minimum point. Intercepts are points where the graph crosses the y-axis. The degree and leading coefficient of a polynomial function determine its end behavior, resulting in smooth curves without breaks and indicating the presence of single or double roots.
Understanding these characteristics of a polynomial function is essential for interpreting graphs and solving equations involving polynomial functions. They provide valuable information about the behavior and properties of the function, enabling a deeper understanding of algebraic expressions and their graphical representations.