3.2 Polynomial function

A real polynomial, simply referred as polynomial in our study, is an algebraic expression having terms of “x” raised to non-negative numbers, separated by “+” or “-“ sign. A polynomial in one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial. A real polynomial function in one variable is an algebraic expression having terms of real variable “x” raised to non-negative numbers. The general form of representation is :

$$f\left(x\right)=a_o+a_{1}x+a_2{x}^2+\dots \dots +a_n{x}^n$$

or

$$f\left(x\right)=a_o{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots \dots +a_n$$

Here, $a_o$ , $a_1$ ,…., $a_n$ are real numbers. For real function, “x” is real variable and “n” is a non-negative number. An expression like $2x^2+2$ is a valid polynomial in “x”. But, $x+1/x$ is not as $1/x=x^{-1}$ has negative integer power. Also, $3x^{1.2}+2x$ is not a polynomial as it contains a term with fractional power. Sum and difference of two real polynomials is also a polynomial. Polynomials are continuous function. Its domain is real number set R, whereas its range is either real number set R or its subset. Derivative and anti-derivative (indefinite integral) of a polynomial are also real polynomials.

Degree of polynomial function/ expression

Highest power in the expression is the degree of the polynomial. The degree of the polynomial $x^3+x^2+3$ is 3. The degree “1” corresponds to linear, degree “2” to quadratic, “3” to cubic and “4” to bi-quadratic polynomial. The general form of quadratic equation is :

$$a{x}^2+bx+c;a,b,c\in R;a\ne 0$$

Note that “a” can not be zero because degree of function/ expression reduces to 1. Extending this requirement for maintaining order of polynomial, we define polynomial of order “n” as :

$$f\left(x\right)=a_o{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots \dots +a_n;a_0\ne 0$$

Polynomial equation

The polynomial equation is formed by equating polynomial to zero.

$$f\left(x\right)=a_o{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots \dots +a_n=0$$

A quadratic equation has the form :

$$f\left(x\right)=a{x}^2+bx+c=0$$

The roots of a polynomial equation are the values of “x” for which value of polynomial f(x) becomes zero. If f(a) = 0, then "x=a" is the root of the polynomial. A polynomial equation of degree “n” has at the most “n” roots – real or imaginary. Important point to underline here is that a real polynomial can have imaginary roots.

Solution of polynomial equation is intersection(s) of two equations :

$$y=a_o{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots \dots +a_n=0$$

and

$$y=0\text{(x-axis)}$$

The solutions of equations (real or complex) are the roots of the polynomial equation. If we plot y=f(x) .vs. y=0 plot, then real roots are x-coordinates (x-intercepts) where plot intersect x-axis. Clearly, graph of polynomial can at most intersect x-axis at “n” points, where “n” is the degree of polynomial. On the other hand, y-intercept of a polynomial is obtained by putting x=0,

$$y=a_{0}X0+a_{1}X0+a_{2}X0+\dots \dots +a_n=a_n$$

Polynomial equation

Some useful deductions about roots of a polynomial equation and their nature are :

1 : A polynomial equation of order n can have n roots – real or imaginary.

2 : Imaginary roots occur in pairs like 1+3i and 1-3i