Polynomial Function

Functions are the base of Calculus. There are so many types of functions that to understand a polynomial function, one must understand what polynomials are. Let us recall the definition of a polynomial.

Definition : 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:

\(a_n x^n~+~a_{n-1} x^{n-1}~+.……….…+~a_2 x^2~+~a_1 x~+~a_0\)

where all the powers are non-negative integers.

\(a_0~,~a_1~,………,~a_n~∈~R.\)

A polynomial is called a univariate or multivariate if number of variables is one or more respectively. So, the variables of a polynomial can have only positive powers. A polynomial function is a function that can be expressed in form of a polynomial. The definition can be derived from the definition of a polynomial. 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 behavior of the function P(x) when x becomes very large. The domain of a polynomial function is entire real numbers (R).

If P(x) = \(a_n x^n~+~a_{n-1} x^{n-1}~+.……….…+~a_2 x^2~+~a_1 x~+~a_0\), then for x ≫ 0 or x ≪ 0, P(x) ≈ \(a_n x^n\). Thus, polynomial functions approach power functions for very large values of their variables.

Polynomial Function Graph with Examples

The graph of P(x) depends upon its degree.

A polynomial having one variable which has the largest exponent is called as degree of the polynomial.

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

Zero Polynomial

    • Degree 0 (Constant Functions) – Standard form: P(x) = a = \( a.x^0\), 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 fig. 1)

Polynomial function

Figure 1: y = 4

Linear Polynomial

  • 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 represent the y-intercept of a line.

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

here a = 2 and b = 3

Polynomial function

Figure 2: y = 2x + 3

Note: All constant functions are linear functions.

  • Degree 2, Quadratic Functions – Standard form: P(x) = \(ax^2~+~bx~+~c\) , where a, b and c are constants

    Graph: A parabola is a curve with one extreme point called as vertex. A parabola is 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) = \( ~\left( – \frac {b}{2a}~,~- \frac{D}{4a} \right) \)

where D is the discriminant given by (\(b^2~-~4ac\))

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 = \( x^2~+~2x~-~3\)  (shown in black color)

y = \( -x^2~-~2x~+~3\) (shown in blue color)

See fig. 3

Polynomial function

Figure 3: y = \(x^2~+~2x~-~3~ (black)~ and ~y\)= \(-x^2~-~2x~+~3~(blue)\)

  • Higher degree: Standard form  P(x) = \(a_n x^n~+~a_{n-1} x^{n-1}~+.……….+a_0\), where \(a_0~,~a_1,………,a_n\) 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. \(a_0\)  here represents the y-intercept.

E.g. y = \(x^4~-~2x^2~+~x~-~2\) , any straight line can intersect it at maximum of 4 points (see fig. 4)

Polynomial function

Figure 4: y = \( x^4~-~2x^2~+x~-~2\)

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Practise This Question

If I < x < I +1, Find [-x], where I is an integr