Table of Contents

## Introduction to Polynomials

Polynomials are defined as the type of algebraic expressions whose variables have only non-negative integers as their powers.

For example:

f(x) = 5x² – x + 1 is a polynomial.

The algebraic expression 3x³ + 4x + 5/x + 6x³/² is not a polynomial since one of the powers of ‘x’ is a fraction and the other is negative.

- Algebraic expressions with the power in either negative or fraction aren’t considered Polynomials.
- The terms in the polynomial expression comprise the variables, exponents, and constants.
- The first term of the polynomial expression in the standard form is called the “leading term.”

A **standard polynomial** is a polynomial whose first term is the highest power holder, and the subsequent terms are arranged in descending order of the powers.

The number multiplied by a variable is called the “**coefficient**.” The number without any variable is called a “**constant**.”

### Expression of the standard polynomial

a^{xm} + bx^{(m-1)} + bx^{(m-2)} + ……… + nx⁰ is the standard polynomial.

Where x⁰ equals 1, therefore, the coefficient is a constant.

In this article, we will discuss the properties of the polynomial functions.

### Definition of Polynomials

Polynomials are algebraic expressions that consist of a combination of variables and coefficients. Variables are also referred to as indeterminates.

We can perform a number of arithmetic operations on polynomials, such as addition, subtraction, multiplication, and more.

An example of a polynomial with one variable is x²+x-1. In this example, there are three terms: x², x, and -1.

### What is the meaning of Polynomial?

The word polynomial is derived from the Greek words ‘poly,’ which means ‘many,’ and ‘nominal,’ which means ‘terms,’ so altogether, it means “many terms.” A polynomial can have any number of terms but can never have an infinite number of terms.

### Notation of Polynomials

The polynomial function is denoted by P(x), where x represents the variable.

For example, P(x) = x²-5x+9

If the variable is represented by a, then the function will be P(a).

## Properties of the Polynomials

Some of the important properties of polynomials are mentioned below.

**Property 1:**

### Division Algorithm

If a polynomial f(x) is divided by a polynomial g(x), then as result it gives us quotient Q(x) with remainder R(x), then,

f(x) = g(x) • Q(x) + R(x)

Where,

R(x)=0 or the degree of R(x) is lesser than the degree of g(x)

**Property 2:**

### Bezout’s Theorem

Bezout’s theorem states that a polynomial f(x) is divisible by a binomial (x – a) iff f(a) = 0.

**Property 3:**

### Remainder Theorem

The Remainder theorem states that if P(x) is divided by (x – a) with remainder r, then P(a) = r.

**Property 4:**

### Factor Theorem

A polynomial f(x) divided by q(x) results in r(x) with zero remainder iff q(x) is a factor of f(x).

**Property 5:**

### Intermediate Value Theorem

If f(x) is a polynomial, and f(x) ≠ f(y) for (x < y), then f(x) takes every value from f(x) to f(y) in the closed interval [x, y].

**Property 6**

The addition, subtraction, and multiplication arithmetic operation on the polynomials P and Q, we get the resulting polynomial whose,

Degree(P ± Q) ≤ Degree(P or Q)

Degree(P × Q) = Degree(P) + Degree(Q)

**Property 7**

If a polynomial f(x) is divisible by a polynomial g(x), then every zero of g(x) is also a zero of f(x).

**Property 8**

If a polynomial f(x) is divisible by two coprime polynomials g(x) and r(x), then it is divisible by (g(x) • r(x)).

**Property 9**

If f(x) = a0 + a1x + a0x² + …… + anxn is a polynomial such that deg(f(x)) = n ≥ 0, then f has at most “n” distinct roots.

**Property 10: **

### Fundamental Theorems of Algebra

Every non-constant single-variable polynomial with complex coefficients always has at least one complex zero.

**Property 11**

If f(x) is a polynomial with real coefficients and has one complex zero, say x = a – bi, then x = a + bi will also be a zero of f(x).

Also check the **Types of polynomials**

## Frequently Asked Questions on Polynomials

### What is a Polynomial?

A polynomial is an expression consisting of variables (or indeterminate), exponents, and constants. For example, 3x2 -2x-10 is a polynomial.

### What is the standard form of the polynomial?

A standard polynomial is one where the highest degree is the first term; subsequently, the other terms come. For example, x3 – 3x2 + x -12 is a standard polynomial. So the highest degree in the given polynomial expression is 3, then comes 2, and then 1.

### Is 6 a polynomial?

6 can be written as 6x0 or 0x2+0x+6, representing the polynomial expression. Therefore, we can consider 6 as a polynomial.