The degree of a polynomial is defined as the highest exponent in a polynomial, within any one term. In a polynomial, each term is of the form [tex]\( ax^n \)[/tex], where [tex]\( a \)[/tex] is a coefficient (which could be a number or another expression), [tex]\( n \)[/tex] is the exponent, and [tex]\( x \)[/tex] is the variable. The degree gives us information about the behavior of the polynomial at extremely large or small values of [tex]\( x \)[/tex].
Let's review the options given to further clarify the correct response:
- The highest exponent in a polynomial (in any one term): This is the correct definition of the degree of a polynomial. For example, in [tex]\( 2x^3 + 3x^2 + x + 5 \)[/tex], the highest exponent is 3, so the degree of this polynomial is 3.
- A base with an exponent: This seems to describe a term with an exponent, but it is not the definition of the degree of a polynomial. An individual base with an exponent is merely part of a term within a polynomial.
- A number that tells how many powers there are: This is a vague statement and could be misinterpreted in many ways. It doesn't directly provide the correct definition of the degree of a polynomial.
- The number of times the base occurs: This option is incorrect because the degree is not determined by how many times a base occurs. It is determined by the largest exponent on the variable within any term of the polynomial.
- A number or variable that has an exponent: This is also not the definition of the degree of a polynomial. This description might refer to any component of a polynomial that contains an exponent, but it doesn't capture the essence of what the degree is.
Therefore, the correct response is the first option: "The highest exponent in a polynomial (in any one term)." This is the standard definition used in mathematics to describe the degree of a polynomial.
