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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 x = a o + a 1 x + a 2 x 2 + + a n x n

or

f x = a o x n + a 1 x n 1 + a 2 x n 2 + + 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 2 x 2 + 2 is a valid polynomial in “x”. But, x + 1 / x is not as 1 / x = x - 1 has negative integer power. Also, 3 x 1.2 + 2 x 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 + b x + c ; a , b , c R ; a 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 x = a o x n + a 1 x n 1 + a 2 x n 2 + + a n ; a 0 0

Polynomial equation

The polynomial equation is formed by equating polynomial to zero.

f x = a o x n + a 1 x n 1 + a 2 x n 2 + + a n = 0

A quadratic equation has the form :

f x = a x 2 + b x + 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 + + a n = 0

and

y = 0 (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 X 0 + a 1 X 0 + a 2 X 0 + + a n = a n

X and y intercepts of polynomial

Graph of polynomial can at most intersect x-axis at “n” points, where “n” is the degree of polynomial.

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

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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