Polynomial Functions- Definition, Formula, Types and Graph With Examples (2024)

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, 2x+5 is a polynomial that has exponent equal to 1.

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A polynomial function, in general, is also stated as a polynomial or polynomial expression, defined by its degree. The degree of any polynomial is the highest power present in it. In this article, you will learn polynomial function along with its expression and graphical representation of zero degrees, one degree, two degrees and higher degree polynomials.

Table of Contents:

  • Definition
  • Examples
  • Types
  • Graphs
  • Questions

Polynomial Function Definition

A polynomial function is a function that can be expressed in the form of a polynomial. The definition can be derived from the definition of a polynomial equation. A polynomial is generally represented as P(x). The highest power of the variable of P(x) is known as its degree. Degree of a polynomial function is very important as it tells us about the behaviour of the function P(x) when x becomes very large. The domain of a polynomial function is entire real numbers (R).

If P(x) = an xn + an-1 xn-1+.……….…+a2 x2 + a1 x + a0, then for x ≫ 0 or x ≪ 0, P(x) ≈ an xn. Thus, polynomial functions approach power functions for very large values of their variables.

Polynomial Function Examples

A polynomial function has only positive integers as exponents. We can even perform different types of arithmetic operations for such functions like addition, subtraction, multiplication and division.

Some of the examples of polynomial functions are here:

  • x2+2x+1
  • 3x-7
  • 7x3+x2-2

All three expressions above are polynomial since all of the variables have positive integer exponents. But expressions like;

  • 5x-1+1
  • 4x1/2+3x+1
  • (9x +1) ÷ (x)

are not polynomials, we cannot consider negative integer exponents or fraction exponent or division here.

Also, see:

  • Polynomial Formula
  • Multiplying Polynomials
  • Polynomial Class 10 Notes
  • Polynomial Class 9 Notes

Types of Polynomial Functions

There are various types of polynomial functions based on the degree of the polynomial. The most common types are:

  • Constant Polynomial Function: P(x) = a = ax0
  • Zero Polynomial Function: P(x) = 0; where all ai’s are zero, i = 0, 1, 2, 3, …, n.
  • Linear Polynomial Function: P(x) = ax + b
  • Quadratic Polynomial Function: P(x) = ax2+bx+c
  • Cubic Polynomial Function: ax3+bx2+cx+d
  • Quartic Polynomial Function: ax4+bx3+cx2+dx+e

The details of these polynomial functions along with their graphs are explained below.

Graphs of Polynomial Functions

The graph of P(x) depends upon its degree. A polynomial having one variable which has the largest exponent is called a degree of the polynomial.

Let us look at P(x) with different degrees.

Constant Polynomial Function

Degree 0 (Constant Functions)

  • Standard form: P(x) = a = a.x0, where a is a constant.
  • Graph: A horizontal line indicates that the output of the function is constant. It doesn’t depend on the input.

E.g. y = 4, (see Figure 1)

Figure 1: y = 4

Zero Polynomial Function

A constant polynomial function whose value is zero. In other words, zero polynomial function maps every real number to zero, f: R → {0} defined by f(x) = 0 ∀ x ∈ R. For example, let f be an additive inverse function, that is, f(x) = x + ( – x) is zero polynomial function.

Linear Polynomial Functions

Degree 1, Linear Functions

  • Standard form: P(x) = ax + b, where a and b are constants. It forms a straight line.
  • Graph: Linear functions have one dependent variable and one independent which are x and y, respectively.

In the standard formula for degree 1, a represents the slope of a line, the constant b represents the y-intercept of a line.

E.g., y = 2x+3(see Figure 2)

here a = 2 and b = 3

Polynomial Functions- Definition, Formula, Types and Graph With Examples (2)

Figure 2: Graph of Linear Polynomial Functions

Figure 2: y = 2x + 3

Note: All constant functions are linear functions.

Quadratic Polynomial Functions

Degree 2, Quadratic Functions

  • Standard form: P(x) = ax2+bx+c , where a, b and c are constant.
  • Graph: A parabola is a curve with one extreme point called the vertex. A parabola is a mirror-symmetric curve where any point is at an equal distance from a fixed point known as Focus.

In the standard form, the constant ‘a’ represents the wideness of the parabola. As ‘a’ decreases, the wideness of the parabola increases. This can be visualized by considering the boundary case when a=0, the parabola becomes a straight line. The constant c represents the y-intercept of the parabola. The vertex of the parabola is given by

(h,k) = (-b/2a, -D/4a)

where D is the discriminant and is equal to (b2-4ac).

Note: Whether the parabola is facing upwards or downwards, depends on the nature of a.

  • If a > 0, the parabola faces upward.
  • If a < 0, the parabola faces downwards.

E.g. y = x2+2x-3 (shown in black color)

y = -x2-2x+3 (shown in blue color)

(See Figure 3)

Polynomial Functions- Definition, Formula, Types and Graph With Examples (3)

Figure 3: Quadratic Polynomial Functions

Figure 3: y = x2+2x-3 (black) and y = x2-2x+3 (blue)

Graphs of Higher Degree Polynomial Functions

  • Standard form P(x) = an xn + an-1 xn-1+.……….…+ a0, where a0,a1,………,an are all constants.
  • Graph: Depends on the degree, if P(x) has degree n, then any straight line can intersect it at a maximum of n points. The constant term in the polynomial expression, i.e. a0 here represents the y-intercept.
  • E.g. y = x4-2x2+x-2, any straight line can intersect it at a maximum of 4 points (see fig. 4)

Polynomial Functions- Definition, Formula, Types and Graph With Examples (4)

Figure 4: Graphs of Higher Degree Polynomial Functions

Video Lesson on Quadratic Equations

Polynomial Functions- Definition, Formula, Types and Graph With Examples (5)

Polynomial Function Questions

Q.1: What is a Polynomial?

A polynomial is defined as an expression formed by the sum of powers of one or more variables multiplied to coefficients. In its standard form, it is represented as:
an xn + an-1 xn-1+.……….…+a2 x2 + a1 x + a0

where all the powers are non-negative integers.

And, a0,a1,………,an ∈ R

A polynomial is called a univariate or multivariate if the number of variables is one or more, respectively. So, the variables of a polynomial can have only positive powers.

Q.2: What is the Degree of Polynomial?

The degree of any polynomial expression is the highest power of the variable present in its expression. Constant (non-zero) polynomials, linear polynomials, quadratic, cubic and quartics are polynomials of degree 0, 1, 2, 3 and 4 , respectively. The function f(x) = 0 is also a polynomial, but we say that its degree is ‘undefined’.

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Frequently Asked Questions on Polynomial Functions

Q1

What is meant by a Polynomial function?

A polynomial function is a function that can be expressed in the form of a polynomial. It has a general form of P(x) = anxn + an – 1xn – 1 + … + a2x2 + a1x + ao, where exponent on x is a positive integer and ai’s are real numbers; i = 0, 1, 2, …, n.

Q2

What is a zero polynomial function?

A polynomial function whose all coefficients of the variables and constant terms are zero. In other words, zero polynomial function maps every real number to zero, f: R → {0} defined by f(x) = 0 ∀ x ∈ R.

Q3

What is the domain of a polynomial function?

The domain of a polynomial function is real numbers.

Q4

What does a quadratic polynomial function graphically represent?

A quadratic polynomial function graphically represents a parabola.

Polynomial Functions- Definition, Formula, Types and Graph With Examples (2024)

FAQs

What is the definition of a polynomial function with example? ›

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, 2x+5 is a polynomial that has exponent equal to 1.

What are examples of types of polynomial functions? ›

General Form of Different Types of Polynomial Function
DegreeTypeGeneral Form
1LinearP(x) = px + q
2QuadraticP(x) = px² + qx +r
3CubicP(x) = px³ + qx² + rx + s
4QuadraticP(x) = px⁴ + qx³ + rx² + sx¹ +t
1 more row

What is the formula of a polynomial function? ›

A polynomial is a function of the form f(x) = anxn + an−1xn−1 + ... + a2x2 + a1x + a0 . The degree of a polynomial is the highest power of x in its expression. Constant (non-zero) polynomials, linear polynomials, quadratics, cubics and quartics are polynomials of degree 0, 1, 2 , 3 and 4 respectively.

What is polynomial function and its graph? ›

The graph of a polynomial function will touch the x-axis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities. The higher the multiplicity, the flatter the curve is at the zero. The sum of the multiplicities is the degree of the polynomial function.

What are 5 examples of polynomials? ›

Polynomial
  • 2a + 5b is a polynomial of two terms in two variables a and b.
  • 3xy + 5x + 1 is a polynomial of three terms in two variables x and y.
  • 3y4 + 2y3 + 7y2 – 9y + 3/5 is a polynomial of five terms in two variables x and y.
  • m + 5mn – 7m2n + nm2 + 9 is a polynomial of four terms in two variables m and n.

What is polynomial definition and types? ›

Polynomials are algebraic expressions that may comprise of exponents which are added, subtracted or multiplied. Polynomials are of different types. Namely, Monomial, Binomial, and Trinomial. A monomial is a polynomial with one term. A binomial is a polynomial with two, unlike terms.

What are the four types of polynomial graphs? ›

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.

What is a polynomial function for dummies? ›

In Algebra II, a polynomial function is one in which the coefficients are all real numbers, and the exponents on the variables are all whole numbers. A polynomial whose greatest power is 2 is called a quadratic polynomial; if the highest power is 3, then it's called a cubic polynomial.

What are the basics of polynomials? ›

Polynomials are algebraic expressions that consist of variables and coefficients. Variables are also sometimes called indeterminates. We can perform arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions but not division by variable.

How to calculate a polynomial? ›

1.1: Solve Polynomial Equations by Factoring
  1. Difference of squares:a2−b2=(a−b)(a+b)
  2. Sum of squares: a2+b2 no general formula.
  3. Difference of cubes: a3−b3=(a−b)(a2+ab+b2)
  4. Sum of cubes: a3+b3=(a+b)(a2−ab+b2)
  5. If a binomial is both a difference of squares and a difference cubes, then first factor it as difference of squares.
Sep 13, 2022

What is a real life example of a polynomial graph? ›

Some real world example of polynomial graphs include a U-shape or a roller coaster. They include curves with inflection points.

What are the rules of polynomials? ›

A plain number can also be a polynomial term. In particular, for an expression to be a polynomial term, it must contain no square roots of variables, no fractional or negative powers on the variables, and no variables in the denominators of any fractions.

What is a polynomial equation? ›

A polynomial equation is an equation that has multiple terms made up of numbers and variables. Polynomials can have different exponents. The degree of a polynomial is its highest exponent. The degree tells us how many roots can be found in a polynomial equation.

What do you mean by polynomial with example? ›

In math, a polynomial is a mathematical expression that contains two or more algebraic terms that are added, subtracted, or multiplied (no division allowed!). Polynomial expressions include at least one variable and typically include constants and positive exponents as well. The expression x2 − 4x + 7 is a polynomial.

How do you identify a polynomial? ›

The polynomials can be identified by noting which expressions contain only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The non-polynomial expressions will be the expressions which contain other operations. Explain why the non-polynomial expressions are not polynomials.

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