Polynomial Functions

Polynomial functions can be quadratic, cubic, quartic, etc. with powers that are positive integers. A polynomial is defined by its degree and the number of terms it has.

Polynomial functions are used extensively in maths to represent relations, graphs, etc. Therefore, it is important to understand how they are used and their properties. They help in simplifying and accurately presenting several algebraic expressions. Polynomial functions  are important for anyone who wants to gain a proper understanding of how equations relate to graphs.

What is a Polynomial Function?

A polynomial function is a function with variables and coefficients. Polynomials only have the mathematical operations of addition, subtraction, multiplication, and division. Their exponential powers are only non-negative integers. It will be easier to understand with a real example as follows:

f(x) = 4

The above equation is a polynomial whose highest power is 3, and so its degree is three. It has three terms.

Standard Form of Polynomial Function

A polynomial function can be represented by the following formula:

f(x) = 

There are certain characteristics of the above expression. All the ‘a’s are real numbers and are the coefficients. The first coefficient is called the leading coefficient and its value can never be zero. The power of n is always an integer that is positive. That means the variable can never be in the denominator of a polynomial.

Degree of a Polynomial Function

Let’s take an example of a polynomial function:

In this equation, the highest exponential power is 3, and so the degree of the polynomial is 3.

Distinguishing Between a Polynomial Function and a Non-polynomial Function

In order to understand the differences between a polynomial function and a non-polynomial function, the following equations can be considered:

Between the two equations, the first one is a polynomial function and the second one is not. This is because the second equation has an exponential power that is a fraction. This implies the following things:

• The variable must not be a denominator in the equation.
• The variable cannot have a root sign.

Representing the Polynomial Functions as Graphs

When creating a graph from a polynomial, some random values of x are taken and the corresponding values of y are obtained by using the values of x. The coordinates thus calculated are then plotted on the graph.

Polynomial Function Graphs

If f(x) is a single constant, then the graph is a straight line where the value of y is equal to the constant and the value of x changes in a straight line. If the polynomial function is a linear equation, then the graph is a diagonal straight line. Hence, functions that are constant functions or linear functions have graphs that are straight lines.

A quadratic function is a function with degree 2. The graph of a quadratic equation is always a u-shaped curve or a parabola. It is affected by the value of the constant. If the coefficient of  is increased, then the parabola is narrowed toward the y axis. If the coefficient of   is decreased, then the slope of both arms of the parabola decreases. If the coefficient is a negative number, then the parabola is inverted and follows the same rules as above. But it should be noted that the values of negative numbers are also inverted.

Polynomial Functions and Their Turning Points

In most polynomial functions, the slope of the graph changes directions after certain values are reached. This is called the turning point of the polynomial graph.

• Quadratic equations have one turning point in a parabolic shape.
• Cubic equations have two turning points in a graph that is mostly diagonal.
• A quartic equation shows up to three turning points near the lowest point of the parabola. But some quartics may have fewer turning points.

A polynomial function of degree n will have up to n-1 turning points.

Roots of Polynomial

If (x-2)(x+3)=0, then the roots are 2 and – 3. The function of this equation is represented by f(x) = (x-2)(x+3). The converse of this also holds true that if 2 and – 3 are the roots, then the function must be f(x) = (x-2)(x+3). This also holds for multiple polynomials of multiple degrees. For example if x = 1, x = 2, x = 3, and x = 4, then the function will be

f(x) = (x – 1)(x – 2)(x – 3)(x – 4)

Another fact to consider is that the root can be repeated any number of times. So, if there is a function like

f(x) = 

Then, the root x = 2 has a multiplicity of 2 and x = 4 has a multiplicity of 3. The multiplicity of the root matters because it can be used to determine where the graph line passes. If the multiplicity is even then the graph line touches the x axis at the point (x,0), but if it is odd, then the graph line passes through the x axis.

Conclusion

Polynomial functions are an important tool in the study of mathematics. Some other areas of study in polynomial functions also include how to multiply them.

Some things to note about polynomial functions are that their degree is the highest power of the function, the polynomial functions of the degree zero are constant functions, and polynomial functions with degree 1 are linear functions.

The maximum number of turning points in the graphs of polynomial functions is determined by subtracting 1 from the degree of the function.