Math Help - Polynomials

In this section of Math Help online, we discuss polynomials and polynomial equations.

What is a polynomial? What are polynomials? Definition of a polynomial

A polynomial in x, denoted by P(x), consists of one or more terms such that the terms are either

• an integral constant or

• the product of an integral constant and a positive intergral power of x.

Examples of polynomials in x and equations that are not polynomials in x are shown below:

• 3 x³  +  2 x²  +  1  is a polynomial in x.

• 3 x³  +  2 x ^1/2  +  1  is a not a polynomial in x. (This is because 1/2 is not an integral power of x.)

• 3 x³  +  2 x ^-2  +  1  is a not a polynomial in x. (This is because -2 is not a positive power of x.)

The degree of a monomial is the sum of the exponents of the variables.

Examples of degrees of a monomial are shown below:

• Monomial    3 x³     has degree of 3

• Monomial    3 x³  Y² Z   has degree of 6

This is because x has degree of 3, Y has degree of 2 and z has degree of 1, so the total degree of the monomial is 3+2+1 = 6.

• Monomial    5    has degree of zero

This is because there is no variable shown in the monomial. When there is no variable terms such as x or y shown in the monomial, the variable still exists but has the degree of zero. All variable with degree zero equal one. For example, the variable is     x ⁰   which is equal to 1. In another word, monomial 5 in the example can be written as    5 x ⁰ .

Since   x ⁰  =  1,  5 x ⁰   =   5  and  there is no need to write x ⁰ out.