Polynomial Functions

A polynomial function is a function which involves only non-negative integer powers of a variable in an equation like the quadratic equation, cubic equation, etc. We can give a general definition of a polynomial, and define its degree. In this article, we will learn polynomial function along with its expression and graphical representation of zero degrees, one degree, two degrees and higher degree polynomial equations.

What is a Polynomial Function?

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.

Types of Polynomial Functions

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

  • Zero Polynomial Function: P(x) = a = ax0
  • 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 with Examples

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.

Zero 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)

Graph of Constant Polynomial Functions (Zero Polynomial Functions)

Figure 1: Graph of Zero Polynomial Function

Figure 1: y = 4

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

Graph of Linear Polynomial Functions

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 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’ decrease, 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 given by (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)

Quadratic Polynomial Functions

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)
Graphs of Higher Degree Polynomial Functions

Figure 4: Graphs of Higher Degree Polynomial Functions

Below is a brief recall about polynomials and its degree.

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.

Degree of Polynomial

The degree of any polynomial expression is the highest power of the variable present in its equation. Constant (non-zero) polynomials, linear polynomials, quadratics, cubics 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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