Polynomial Functions

Polynomial Functions are functions consisting of many algebraic terms including constants, variables of different degrees, coefficients, and positive exponents. The degree of the polynomial function is the highest exponent of the variable.

In this article, we will learn about Polynomial Functions, their examples, the degree of polynomial functions, etc. We will also learn about the types of polynomial functions, their graphs, the roots of polynomial functions, and how to identify whether the function is a polynomial function or not.

What is a Polynomial Function?

Polynomial Functions are functions with many terms containing the constant, and variables with different positive exponents and coefficients.

It is generally represented by P(x). Polynomial functions have many terms as in polynomial, poly means many, and nominal means terms. The exponent of the polynomial function must be positive. The domain of the polynomial function is all the real numbers R.

Polynomial Function Definition

The generalized form of the polynomial function is defined as:

P(x) = a_nxⁿ + a_n-1x^n-1 + …… + a₂x² + a₁x + a₀

Where, 

• a_n, a_n-1, . . . a₂, a₁, a₀ are coefficients, 
• x is variable, and
• P(x) is the polynomial function in variable x.

The exponents of the variable should be the whole number. a_n, a_n-1,…… a₂, a₁, a₀ coefficients are real number constants. n is a positive number which is the degree of the polynomial. a_n is the leading coefficient as it is the coefficient of the degree of the polynomial. The leading coefficient in the polynomial function cannot be zero.

Degree of Polynomial Function

The degree of the polynomial function is the highest power of the variable in the function.

Example: Find the Degree of polynomial function P(x) that is given as follows:

P(x) = 4x³ + 3x² + 2x + 1

Solution:

• Degree of term 4x³ is 3, 
• Degree of term 3x² is 2,
• Degree of term 2x is 1, and
• Degree of term 1 is 0.

As the highest degree in all the terms is 3. 

Thus, the degree of the above polynomial function P(x) is 3.

Learn more about the Degree of Polynomials

Examples of Polynomial Function

Some examples of the polynomial functions are:

• Linear Polynomial: P(x)=2x+3
  • This is a first-degree polynomial (linear function) with a slope of 2 and a y-intercept of 3.
• Quadratic Polynomial:P(x)=x²−4x+4
  • This is a second-degree polynomial (quadratic function) representing a parabola.

• Cubic Polynomial: P(x)=x³−3x²+2x

• Quartic Polynomial: P(x)=4x⁴−x³+2x²−5x+1

• Quintic Polynomial: P(x)=x⁵−2x⁴+x−1

• Higher-Degree Polynomial: P(x)=2x⁶−3x⁴+x³−2x+5

• Constant Polynomial: P(x) = 7

• Polynomial with Multiple Variables: P(x,y)=3x²y−2xy²+xy

Types of Polynomial Functions

We can classify the polynomial functions based on various different parameters such as the number of terms it contains or their degree. Classification of polynomial functions based on these parameters is given below:

Based on the Number of Terms

Based on the Number of Terms, a polynomial Function can be classified as follows:

• Monomial
• Binomial
• Trinomial
• and so on …

Let’s learn about these types with their examples in detail as follows:

Monomial: Monomials are polynomial functions with only one term. Some examples of monomials are:

• f = 2xy
• g = 4a²
• h = -3b³c²
• etc.

Binomial: Binomials are polynomial functions with only two terms. Some examples of binomials are:

• f = 3x + 2y
• g = x² + 2xy
• h = 4m² – 6n
• etc.

Trinomial: Trinomials are polynomial functions with only three terms. Some examples of trinomials are:

• f = 2x + 3y – z
• g = x² + 5xy – 2y²
• h = 3m³ + 2m²n – mn²
• etc.

Based on the Degree

Based on the degree of the given polynomial function we can classify the function into the following types:

• Constant Polynomial Function: The polynomial function with the degree zero i.e., only constant is called the constant polynomial function. It is of the form P(x) = a.

• Zero Polynomial Function: The polynomial function which is zero is called the zero-polynomial function. It is of the form P(x) = 0.

• Linear Polynomial Function: The polynomial function with the degree one is called the linear polynomial function. It is of the form P(x) = ax + b.

• Quadratic Polynomial Function: The polynomial function with degree two is called the quadratic polynomial function. It is of the form P(x) = ax² + bx + c.

• Cubic Polynomial Function: The polynomial function with the degree three is called the cubic polynomial function. It is of the form P(x) = ax³ + bx² + cx + d.

• Quartic Polynomial Function: The polynomial function with the degree one is called the quartic polynomial function. It is of the form P(x) = ax⁴ + bx³ + cx² + dx + e.

Learn more about Types of Polynomials

Polynomial Functions Graphs

Polynomial Functions are graphed in many ways depending on the degree of the given polynomial function. Some polynomial functions are graphed as a line, some as parabolas, and some higher-degree polynomial functions are graphed as curves intersecting the x-axis various times.

Let’s understand these graphs individually in detail.

Graph for Constant Polynomial Function

The constant polynomial function is of the form P(x) = a. In the graph, a horizontal line represents the constant polynomial function.

Graph for Linear Polynomial Function

The linear polynomial function is of the form P(x) = ax + b. In the graph, a straight line (with slope a and intercept b) represents the linear polynomial function.

Graph for Quadratic Polynomial Function

The quadratic polynomial function is of the form P(x) = ax² + bx + c. In the graph, a parabola represents the quadratic polynomial function.

Graph for Higher Polynomial Function

The higher polynomial function is of the form P(x) = a_nxⁿ + a_n-1x^n-1 + …… + a₂x² + a₁x + a₀. In the graph, a straight line can intersect the graph on n points if the polynomial function is of degree n.

Zeros of Polynomial Function

Zeroes of the polynomial functions are the numbers that satisfy the equation P(x) = 0. Zeros are also called as the zeros of the polynomial function or the intercepts of the polynomial function.

We have to put P(x) = 0 and solve the equation to obtain the required roots of the polynomial function P(x).

Polynomial function	Maximum Number of roots
Linear	1
Quadratic	2
Cubic	3
Quartic	4

Learn more about how to calculate the Zeros of Polynomials

How to Identify a Polynomial Function?

To check whether a given function is polynomial or not, there are some Rules, that are given as follows:

• The exponent of the function should be a positive number. It should not be negative or fractional.
• The variable should not be radical.
• The denominator should not contain any variable.

Solved Problems of Polynomial Function

Let’s solve some problems on Polynomial functions.

Problem 1: Identify whether the function is a polynomial function or not.

• Q(x) = 5x^-9 + 2
• P(x) = 2x^1/2 + 3
• F(x) = 4x³ + 7

Solution:

• Q(x) = 5x^-9 + 2 is not a polynomial function as it has negative exponent.
• P(x) = 2x^1/2 + 3 is not a polynomial function as it has fractional exponent.
• F(x) = 4x³ + 7 is a polynomial function as it has positive exponent.

Problem 2: Find the zeros of the polynomial function, P(x) = 3x – 15.

Solution:

To find the roots for the polynomial function we have to equate it to 0.

P(x) = 0

⇒ 3x – 15 = 0

⇒ 3x = 15

⇒ x = 5

The root of polynomial function P(x) = 3x -15 is 5.

Problem 3: Find the zeros of the polynomial function, P(x) = x² + 5x + 6.

Solution:

To find the roots for the polynomial function we have to equate it to 0.

P(x) = 0

x² + 5x + 6 = 0

⇒ x² + 3x + 2x + 6 = 0

⇒ x(x+3) + 2(x + 3) = 0

⇒ (x + 3) (x + 2) = 0

⇒ x = -2, -3

The roots of polynomial function P(x) = x² + 5x + 6 are -3 and -2.

Problem 4: Find the zeros of the polynomial function, P(x) = x³ – 5x² – x + 5.

Solution:

To find the roots for the polynomial function we have to equate it to 0.

P(x) = 0

x³ – 5x² – x + 5 = 0

⇒ x²(x – 5) -1(x – 5) = 0

⇒ (x² – 1) (x – 5) = 0

⇒ (x – 1) (x + 1) (x – 5) = 0

⇒ x = 1, -1, 5

The roots of polynomial function P(x) = x³ – 5x² – x + 5 are 1, -1, 5.

FAQs on Polynomial Function

What is Polynomial Function?

The polynomial function is the function consisting of various terms including constants, and variables with different positive exponents and coefficients. It is represented as P(x).

What is the Degree of Polynomial Function?

The degree of a polynomial function is the highest power of the variable.

State various types of Polynomial Functions based on the Degree of Polynomial Function.

Various types of polynomial functions based on the degree of polynomial function are:

• Zero Polynomial Function
• Constant Polynomial Function
• Linear Polynomial Function
• Quadratic Polynomial Function
• Cubic Polynomial Function
• Quartic Polynomial Function

Which Curve Does the Graph of Quadratic Polynomial Function Represent?

The graph of quadratic functions represents the curve parabola.

Write the generalized or standard form of the polynomial function.

The generalized form of the polynomial function is:

P(x) = a_nxⁿ + a_n-1x^n-1 + …… + a₂x² + a₁x + a₀

Where, 

• a_n, a_n-1, . . . a₂, a₁, a₀ are coefficients, 
• x is variable, and
• P(x) is the polynomial function in variable x.

What is Cubic Polynomial Function?

The polynomial function with degree 3 is called cubic polynomial function.