Determine rather the expression is a polynomial.

Question

If it is, state how many terms and variables the polynomial contains. Than state its degree.

Answer 1

A polynomial is a sum of terms where the terms must satisfy the following

1. A constant is a valid term.

2. Other terms must be a constant times a product of powers of variables.

3. The exponents must be positive integer constants.

For example 2, 3x²y³, and -7xz are all valid terms.

Examples of terms that are not allowed:

6x^-5y ← Violates condition 3 because the exponent -5 is negative.

4x^2/3 ← Violates condition 3 because the exponent 2/3 is not an integer.

x^y or 2^y ← Violates condition 3 because the exponent y is not a constant.

Here are a few examples of polynomials:

x²

5xy² - 6z³

x² + y²

etc.

Note: If a term is subtracted then that is the same as adding a term with the opposite sign.

The degree of a term is the sum of the exponents. Here, if a variable is written without an exponent, it's exponent is 1. A constant term's degree is 0.

For example 2xy has degree 1+1 = 2. and x³y⁷z² has degree 3+7+2 = 12.

A polynomials degree is just the largest of the degrees of the terms.

For example, x³ + y²z⁴ + 3 has three terms with respective degrees 3, 6, 0.

Thus this polynomial's degree is the largest of these three values, which is 6.