In mathematics, polynomials are one of the primary structural units when building Math expressions. Let's start by discussing what a monomial is:
Monomial
A monomial is a mathematical expression that consists of two things:
- A constant coefficient \(a\) and
- A variable \(x\)
If we have an expression like \(5x\), the coefficient is five, and the variable is \(x\). It's straightforward as that.
A monomial would have at least a degree – the sum of the exponents of the variables. In the expression \(5x\), the degree is equal to one (since the variable's exponent is one). If we have something like \(2x^2y\), the degree is three (2+1=3).
Polynomial
When we perform a series of math operations among monomials, we have a polynomial.
The expression \(5x^4+2x^2y\) is a polynomial. From this example, we can observe the following:
- The expression consists of two monomials: \(5x^4\) and \(2x^2y\)
- The first one, \(5x^4\), has a coefficient of 5 and a degree of four, while the second one, \(2x^2y\), has a coefficient of 2 and a degree of three.
Most references call the monomials in polynomials "terms." From this point forward, we will refer to these as such.
All polynomial expressions would have the degree of the polynomial \(n\). It is equal to the term with the highest degree. In the expression: \(5x^4+2x^2y\)
- The degree of the polynomial \(n\) is four since the term with the highest degree equals it.
General Form
From these illustrations, we can express every polynomial in a general form:
\(a_n x^n+a_{n-1} x^{n-1}+\cdots+a_2 x^2+a_1 x+a_0\)
- \(a_n\), \(a_{n-1}\), and so forth are the coefficients of the polynomial
- \(n\) is the degree of the polynomial
- \(a_n\) is the leading coefficient of the polynomial - the constant of the highest degree term
- \(a_0\) is the constant coefficient of the polynomial - the constant of the term without a variable
Summary
A monomial is a mathematical expression that consists of two things: a constant coefficient \(a\) and variable \(x\)
A monomial would have at least a degree – the sum of the exponents of the variables.
When we perform a series of math operations among monomials, we have a polynomial.
Most references call the monomials in polynomials "terms."
All polynomial expressions would have the degree of the polynomial \(n\).
We can express every polynomial in a general form: \(a_n x^n+a_{n-1} x^{n-1}+\cdots+a_2 x^2+a_1 x+a_0\)
