Question

what is polynomial function ??​

Answer 1

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A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, involving only non-negative integer powers of x. We can give a general defintion of a polynomial, and define its degree.

Answer 2

Answer:

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.

Explanation:

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.

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)

Graph of Constant Polynomial Functions (Zero Polynomial Functions)

Figure 1: Graph of Zero Polynomial Function

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

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 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.

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