# Polynomial

An expression of the form a$$_{0}$$x$$^{n}$$ + a$$_{1}$$x$$^{n - 1}$$ + a$$_{2}$$x$$^{n - 2}$$ + a$$_{3}$$x$$^{n - 3}$$ + ..... + a$$_{n}$$ where a$$_{0}$$, a$$_{1}$$, a$$_{2}$$, a$$_{3}$$, ....., a$$_{n}$$ are given numbers (real or complex), n is a non-negative integer and x is a variable is called a polynomial in x.

a$$_{0}$$, a$$_{1}$$, a$$_{2}$$, a$$_{3}$$, etc., are called the coefficients of x$$^{n}$$, x$$^{n - 1}$$, x$$^{n - 2}$$, x$$^{n - 3}$$, etc., respectively.

a$$_{0}$$x$$^{n}$$, a$$_{1}$$x$$^{n - 1}$$, a$$_{2}$$x$$^{n - 2}$$, a$$_{3}$$x$$^{n - 3}$$, ....., a$$_{n}$$ are called the terms of the polynomial.

a$$_{n}$$ is called the constant term. Clearly, it is also the coefficient of x$$^{0}$$.

If a$$_{0}$$ ≠ 0, the polynomial is said to be of degree n and the term a$$_{0}$$x$$^{n}$$ is called the leading term.

The general form of a polynomial of degree 1 is a$$_{0}$$x + a$$_{1}$$where a$$_{0}$$ ≠ 0.

The general form of a polynomial of degree 2 is a$$_{0}$$x$$^{2}$$ + a$$_{1}$$x + a$$_{2}$$ where a$$_{0}$$ ≠ 0.

A non-zero constant a$$_{0}$$ itself is said to be a polynomial of degree 0 while a polynomial all of whose coefficients are zero is said to be a zero polynomial and is denoted by 0 and no degree is assigned to it.

Since a polynomial is an expression containing the variable x, it is denoted by f(x), p(x) or g(x) etc.

The value of a polynomial f(x) for x = a where a is real number or a complex number is denoted by f(a).

In particular, if the coefficients a$$_{0}$$, a$$_{1}$$, a$$_{2}$$, a$$_{3}$$, .... of a polynomial f(x) be all real numbers, the polynomial f(x) is said to be a real polynomial.

Examples of polynomial:

(i) 7x$$^{2}$$ + 5x - 3 is a polynomial in x of degree 2 or a quadratic polynomial in x.

(ii) 4x$$^{3}$$ + 9x$$^{2}$$ - 4x + 2 is a polynomial in x of degree 3 or a cubic polynomial in x.

(iii) 5 - 2x$$^{\frac{5}{3}}$$ + 9x$$^{2}$$ is an expression but not a polynomial, since it contains a term containing x$$^{\frac{5}{3}}$$ , where $$\frac{5}{3}$$ is not a non-negative integer.

● Factorization