Overview
Polynomials contain variables, constants, and exponents. The simplest polynomial is called a monomial, as a monomial consists of a single expression. Other polynomials are a sum of monomials.
Monomials
A monomial expression can consist of a constant, variable, and exponent, all multiplied together. The variable is not in the denominator of the expression, such as 1/x, and the exponent must be a whole number, neither a fractional exponent, nor a negative exponent. Therefore, 2x2 is a monomial, but 3x-5 is not, nor is 5y2/3.
Polynomials
A polynomial is a monomial or the sum of monomials. For example, 2x2 +3x is a polynomial that is formed by the sum of the monomials 2x2 and 3x. Similarly, a polynomial such as x2-4x3 +32x +16, is formed by 4 separate monomials, x2, -4x3, 32x, and 16. An expression such as 3y2 + 9y, 10x – 3, or 3a6-9a3, is also called a binomial because it contains two terms. Similarly, an expression such as 2x2-5x-16 is also called a trinomial because it contains three terms. A constant such as 16 can be thought to contain x0, because any variable with a zero exponent equals 1.
Degree of a Polynomial
The degree of a polynomial is the degree of the term in that polynomial with the highest exponent. In an expression such as x2 – 7x3 +94x +124, the degree is 3, because the highest exponent is in the term -7x3. Similarly, in the expression 18 – 12b9, the degree is 9, because the highest exponent is 9. In an expression such as 16x-5, the degree is 1, because 16x has an exponent of 1. By convention, x1 is the same thing as x. We generally leave out an exponent of 1.
Standard Form
When writing polynomial expressions in standard form, the term with the highest degree is written first, then the next highest degree, then the next highest to that, all the way to the constants. Therefore, an expression such as 20x-9x3 + 2 +18x2 can be rearranged in standard form as -9x3 + 18x2 +20x +2, as -9x3 has degree 3 , 18x2 has degree 2, 20x has degree 1, and 2 has degree 0. The coefficient -9 in the term -9x3 is often called the leading coefficient, as it is the coefficient of the highest degree term and “leads” the polynomial when it is rearranged in standard form. If the polynomial were x2 +2x +3, the leading coefficient would be 1. When there is no constant beside a variable in a monomial it is understood to be 1, as 1 times x2 is still x2, using the multiplicative identity.
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