Polynomial vs Monomial
A polynomial is defined as a mathematical expression given as a sum of terms created by products of variables and coefficients. If the expression involves one variable, the polynomial is known as univariate, and if the expression involves two or more variables, it is multivariate.
A univariate polynomial often symbolized as P(x) is given by;
P(x) = an xn + an-1 xn-1 + an-2 xn-2 +⋯+ a0; where, x, a0, a1, a2, a3, a4, … an ∈ R and n ∈ Z0+
[For an expression to be a polynomial, its variable should be a real variable and the coefficient is also real. And the exponents must be non-negative integer]
Polynomials are often distinguished by the highest power of the terms in the polynomial when it is in canonical form, which is called the degree (or order) of the polynomial. If the highest power of any term is n, it is known as an nth degree polynomial [for example, If n=2, it is a second order polynomial; if n=3, it is a 3rd order polynomial].
Polynomial functions are functions where the domain-co-domain relation is given by a polynomial. A quadratic function is a second order polynomial function. Polynomial equation is an equation where two or more polynomials are equated [if the equation is like P = Q, both P and Q are polynomials]. They are also called algebraic equations.
A single term of the polynomial is a monomial. In other words, a summand of a polynomial can be considered as a monomial. It has the form an xn. An expression with two monomials is known as a binomial, and with three terms is known as a trinomial [binomials ⇒ an xn + bn yn, trinomial ⇒ an xn + bn yn + cn zn].
Polynomial are a special case of the mathematical expression and has a wide range of important properties. Sum of polynomials is a polynomial. Product of polynomials is a polynomial. Composition of a polynomial is a polynomial. The differentiation of polynomials produces polynomials.
Also, polynomials can be used to approximate other functions using special methods such as Taylor’s series. For example sin x, cos x, ex can be approximated using polynomial functions. In the field of statistics, the relationships between variable are approximated using polynomials by finding the best fitting polynomial and determining appropriate coefficients.
The quotient of two polynomials produces a rational function (x)=[P(x)] / [Q(x)] , where Q(x)≠0.
Interchanging the coefficients such that a0 ⇌ an, a1 ⇌ an-1, a2 ⇌ an-2, and so on, a polynomial equation, whose roots are the reciprocals of the original, can be obtained.
What is the difference between Polynomial and Monomial?
• A mathematical expression formed by the product of the coefficients and variables and exponentiation of variables is known as a monomial. The exponents are non-negative, and the variables and the coefficients are real.
• A polynomial is a mathematical expression formed by the sum of monomials. Therefore, we can say that monomials are summands of polynomials or a single term of the polynomial is a monomial.
• Monomials cannot have an addition or subtraction among the variables.
• Degree of the polynomials is the degree of the highest monomial.