A polynomial expression is a group of one or more terms that follow these rules:
The variables must have exponents that are whole numbers, and:
The variables cannot have negative exponents.
The variables cannot have fractional exponents.
The variables cannot be in the denominator .
The variables must have no radical or absolute value symbols.
Example 1
Determine if the expression is a polynomial. Explain.
Yes, this is a polynomial because all the variables have exponents that are whole numbers.
No, this is NOT a polynomial because the first term has a negative exponent and the second term has a variable in the denominator.
No, this is NOT a polynomial because the first term has a variable under a radical symbol . Polynomials have whole number exponents . Another way to write a square root is to the one-half power .
Polynomials (cont.) Solutions
If an expression, written in simplest form, is a polynomial, then it can be classified by the degree and number of terms .
The degree of a polynomial is the largest exponent when there is a single variable, or the largest sum of the exponents of the variables in a multivariate polynomial.
A multivariate polynomial is a polynomial with two or more variables.
With multivariate polynomials, add the exponents to find the largest sum even though the bases are not the same.
Degree
Classification
0
constant
1
linear
2
quadratic
3
cubic
4
quartic
5
quintic
n
nth degree
Number of Terms
Classification
1 term
monomial
2 terms
binomial
3 terms
trinomial
n-terms
polynomial with n-terms
Example 2
Classify the polynomial expression by the degree and number of terms.
Expression A is a 4th degree, or quartic polynomial with 4 terms.
1st term exp: 2nd term: 3rd term:
Expression B is a 5th degree trinomial, or quintic trinomial.
1st term: 2nd term: 3rd term:
Expression C is a 9th degree trinomial.
Example 3
Determine whether each expression is a polynomial. If it is a polynomial, state the degree of the polynomial.
No, expression A is not a polynomial because the absolute value of a variable is being taken.
Yes, expression B is a polynomial.
The degree is 9.
Yes, expression C is a polynomial because the 4 is the only value under the radical.
