Fill in the blank(s) to correctly complete each sentence. A polynomial function with leading term 3x^5 has degree ____.

Verified step by step guidance

1. Identify the leading term of the polynomial, which is \(3x^5\).
2. Recognize that the degree of a polynomial is determined by the highest power of the variable \(x\) in the polynomial.
3. Observe that in the leading term \(3x^5\), the exponent of \(x\) is 5.
4. Conclude that the degree of the polynomial is the same as the exponent of the leading term, which is 5.
5. Therefore, the degree of the polynomial function is 5.

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form of a polynomial in one variable is given by f(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It indicates the polynomial's behavior as the variable approaches infinity and plays a crucial role in determining the polynomial's graph and its number of roots. For example, in the polynomial 3x^5 + 2x^3 - x + 7, the degree is 5.

Leading Term

The leading term of a polynomial is the term that contains the highest power of the variable, along with its coefficient. It is significant because it influences the end behavior of the polynomial function. In the polynomial 3x^5, the leading term is 3x^5, which indicates that the polynomial's degree is 5.