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
axm + 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.
