Polynomial Function

A polynomial function is the simplest, most commonly used, and most important mathematical function. These functions represent algebraic expressions with certain conditions. They also cover a wide number of functions. It is essential for one to study and understand polynomial functions due to their extensive applications.

In this article, let's learn about the definition of polynomial functions, their types, and graphs with solved examples.

In this article, we will be learning about the different aspects of polynomial functions. Polynomial is made up of two words, poly, and nomial. "Poly" means many, and "nomial" means the term, and hence when they are combined, we can say that polynomials are "algebraic expressions with many terms". Let’s go ahead and start with the definition of polynomial functions and their types.

Polynomial Function Definition

Polynomial functions are expressions that may contain variables of varying degrees, coefficients, positive exponents, and constants. Here are some examples of polynomial functions.

• f(x) = 3x² - 5
• g(x) = -7x³ + (1/2) x - 7
• h(x) = 3x⁴ + 7x³ - 12x²

Polynomial Function in Standard Form

A polynomial function in standard form is: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x²+ a₁x + a₀. This algebraic expression is called a polynomial function in variable x.

Here,

• a_n, a_n-1, … a₀ are real number constants
• a_n can’t be equal to zero and is called the leading coefficient
• n is a non-negative integer
• Each exponent of variable in polynomial function should be a whole number

Degree of a Polynomial Function

The degree of the polynomial function is the highest power of the variable it is raised to. Consider this polynomial function f(x) = -7x³ + 6x² + 11x – 19, the highest exponent found is 3 from -7x³. This means that the degree of this particular polynomial is 3.

The name of a polynomial is determined by the number of terms in it. The three most common polynomials we usually encounter are monomials, binomials, and trinomials.

• Monomials are polynomials that contain only one term. Examples: 15x², 3b, and 12y⁴
• Binomials are polynomials that contain only two terms. Examples: x + y, 4x – 7, and 9x + 2
• Trinomials are polynomials that contain only three terms. Examples: x³ – 3 + 5x, z⁴ + 45 + 3z, and x² – 12x + 15

Further, the polynomials are also classified based on their degrees. The four most common types of polynomials that are used in precalculus and algebra are zero polynomial function, linear polynomial function, quadratic polynomial function, and cubic polynomial function.

Zero Polynomial Function

A zero polynomial function is of the form f(x) = 0, yes, it just contains just 0 and no other term or variable. Since f(x) = a constant here, it is a constant function.

Linear Polynomial Function

A linear polynomial function has a degree 1. It is of the form f(x) = ax + b. Some examples of a linear polynomial function are f(x) = x + 3, f(x) = 25x + 4, and f(y) = 8y – 3.

Quadratic Polynomial Function

A quadratic polynomial function has a degree 2. It is of the form f(x) = ax² + bx + c. Some examples of a quadratic polynomial function are f(m) = 5m² – 12m + 4, f(x) = 14x² – 6, and f(x) = x² + 4x.

Cubic Polynomial Function

A cubic polynomial function has a degree 3. It is of the form f(x) = ax³ + bx² + cx + d. Some examples of a cubic polynomial function are f(y) = 4y³, f(y) = 15y³ – y² + 10, and f(a) = 3a + a³.

In order to determine if a function is polynomial or not, the function needs to be checked against certain conditions for the exponents of the variables. These conditions are as follows:

• The exponent of the variable in the function in every term must only be a non-negative whole number.
  i.e., the exponent of the variable should not be a fraction or negative number.
• The variable of the function should not be inside a radical i.e, it should not contain any square roots, cube roots, etc.
• The variable should not be in the denominator.

The below-given table shows an example and some non-examples of polynomial functions:

Note: Remember that coefficients can be fractions, negative numbers, 0, or positive numbers. We just need to take care of the exponents of variables to determine whether it is a polynomial function.

We can represent all the polynomial functions in the form of a graph. Remember that the domain of any polynomial function is the set of all real numbers. The below-given image shows the graphs of different polynomial functions. An important skill in coördinate geometry is to recognize the relationship between equations and their graphs.

• A linear polynomial function is of the form y = ax + b and it represents a straight line.
  To know how to graph a linear polynomial function, click here.
• A quadratic polynomial function is of the form y = ax² + bx + c and it represents a parabola.
  To know how to graph a quadratic polynomial function, click here.
• A cubic polynomial function is of the form y = ax³ + bx² + cx + d.
  To know how to graph a cubic polynomial function, click here.

Graphing Polynomial Functions

To graph a simple polynomial function, we usually make a table of values with some random values of x and the corresponding values of f(x). Then we plot the points from the table and join them by a curve. Let us draw the graph for the quadratic polynomial function f(x) = x².

Let's plot the points and join them by a curve (also extend it on both sides) to get the graph of the polynomial function.

If you are curious to know how to graph different types of functions then click here.

The zeros (which are also known as roots or x-intercepts) of a polynomial function f(x) are numbers that satisfy the equation f(x) = 0. So to find the zeros of a polynomial function f(x):

• Set f(x) = 0
• Solve the equation using solving techniques of equations.

Zeros of Linear Polynomial Function

Consider a linear polynomial function f(x) = 16x - 4. To find its zeros:

• Set f(x) = 0
  16x - 4 = 0
• Solve it.
  16x = 4
  x = 1/4

Thus, the zero of f(x) is 1/4.

Zeros of Quadratic Polynomial Function

Consider a quadratic polynomial function f(x) = x² + 2x - 5. To find its zeros:

• Set f(x) = 0.
  Then x² + 2x - 5 = 0.
• Solve it.
  Here a = 1, b = 2 and c = -5.
  Let us use the quadratic formula to find the quadratic roots, x = [-b ± √(b² - 4ac)]/2a
  x = [-2 ± √(2² - 4(1)(-5))]/(2)(1)
  = [-2 ± √(4+20)]/2
  = [-2 ± √(24)]/2
  = [-2 ± 2√6]/2
  = -1 ± √6

Hence, -1 + √6 and -1 -√6 are the zeros of the polynomial function f(x). Remember that the irrational roots and complex roots of a polynomial function always occur in pairs.

Zeros of Cubic Polynomial Function

Finding the zeros of cubic polynomials is same as that of quadratic equations. But to make it to a much simpler form, we can use some of these special products:

• Perfect cube (2 forms): a³ ± 3a²b + 3ab² ± b³ = (a ± b)³
• Difference of the cubes: a³ − b³ = (a − b)(a² + ab + b²)
• Sum of the cubes: a³ + b³ = (a + b)(a² − ab + b²)

Let us find the zeros of the cubic polynomial function f(y) = y³ – 2y² – y + 2.

• Set f(y) = 0.
  y³ – 2y² – y + 2 = 0.
• Solve it.
  y²(y – 2) – (y – 2) = 0
  (y² – 1) (y – 2) = 0
  (y + 1) (y – 1) (y– 2) = 0
  y = 1, -1 and 2.

Hence the zeros of the polynomial function are 1, -1, and 2.

☛Related Articles:

Check out the following pages related to polynomial functions:

• Multiplying Polynomials
• Multiplying Polynomials Calculator
• Multiplying Binomials Calculator
• Polynomial Calculator

Important Notes on Polynomial Functions:

Here is a list of a few points that should be remembered while studying polynomial functions:

• The degree of the polynomial function is determined by the highest power of the variable it is raised to.
• Constant functions are polynomial functions of degree 0.
• Linear Functions are polynomial functions of degree 1.
• Quadratic Functions are polynomial functions of degree 2.
• Cubic Functions are polynomial functions of degree 3.

What Are Polynomial Functions?

Polynomial functions are expressions that are a combination of variables of varying degrees, non-zero coefficients, positive exponents (of variables), and constants. For example, f(b) = 4b² – 6 is a polynomial in 'b' and it is of degree 2.

What Are the Types of Polynomial Functions?

There are various types of polynomial functions that are classified based on their degrees. They are:

• Zero Polynomial Function (f(x) = 0; degree = 0)
• Constant function (f(x) = k; degree = 0)
• Linear Polynomial Function (f(x) = ax + b; degree = 1)
• Quadratic Polynomial Function (f(x) = ax² + bx + c; degree = 2)
• Cubic Polynomial Function (f(x) = ax³ + bx² + cx + d; degree = 3)
• Quartic Polynomial Function (f(x) = ax⁴ + bx³ + cx²+ dx + e; degree = 4)

What is Polynomial Function Formula?

Here is the polynomial function formula: f(x) = a_nxⁿ + a_n-1x^n-1 + ... + a₂x²+ a₁x + a₀. Here,

• a_n, a_n-1, … a₀ are called the coefficients
• a_n is non-zero
• n is a non-negative integer
• Exponents of variables should be non-negative and non-fractional numbers.

How to Find the Degree of Polynomial Functions?

The degree of the polynomial function is determined by the highest power of the variable it is raised to. Consider the polynomial function f(y) = -4y³ + 6y⁴ + 11y – 10, the highest exponent found is 4 from the term 6y⁴. Hence the degree of this particular polynomial is 4.

How to Find the Number of Roots of a Polynomial Function?

The maximum number of roots of a polynomial function is equal to its degree. For example:

• A linear function has one root.
• A quadratic function has a maximum of 2 roots.
• A cubic function has a maximum of 3 roots.

How to Find the Zeros of a Polynomial Function?

The zeros of a polynomial function f(x) are also known as its roots or x-intercepts. Here are the steps to find them:

• Set it equal to zero. i.e., f(x) = 0.
• Solve the equation.

What are the Theorems Related to Polynomial Functions?

Some theorems related to polynomial functions are very helpful in finding their zeros:

• Remainder theorem: It states that the remainder when f(x) is divided by (x - a) is f(a).
• Factor theorem: It states that if (x - a) is a factor of f(x) then f(a) = 0.
• Rational root theorem: A rational root of a polynomial function f(x) is of the form p/q where p is a factor of the constant and q is a factor of the leading coefficient.

What Are the Examples of Polynomial Functions?

Here are a few examples of each type of polynomial function:

• Constant function. Eg: y = 1
• Linear Polynomial Function. Eg: 5y + 10
• Quadratic Polynomial Function. Eg: 14x² + 2x – 6
• Cubic Polynomial Function. Eg: 4y³ + 5y² + 2
• Quartic Polynomial Function. Eg: 3y⁴ + 5