In Exercises 5–8, find the degree of the polynomial. x^2−4x^3+9x−12x^4+63

Verified step by step guidance

1. Identify each term in the polynomial: \(x^2\), \(-4x^3\), \(9x\), \(-12x^4\), and \(63\).
2. Determine the degree of each term by looking at the exponent of the variable \(x\): \(x^2\) has degree 2, \(-4x^3\) has degree 3, \(9x\) has degree 1, \(-12x^4\) has degree 4, and \(63\) has degree 0.
3. The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its standard form.
4. Compare the degrees of all the terms: 2, 3, 1, 4, and 0.
5. Identify the highest degree among these, which is the degree of the polynomial.

Key Concepts

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 overall behavior and the number of roots it can have. For example, in the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63, the term with the highest exponent is -12x^4, making the degree 4.

Polynomial Terms

A polynomial is composed of terms, which are individual components that can include constants, variables, and exponents. Each term is typically in the form of ax^n, where 'a' is a coefficient, 'x' is the variable, and 'n' is a non-negative integer. Understanding how to identify and categorize these terms is essential for determining the degree of the polynomial.

Standard Form of a Polynomial

A polynomial is often expressed in standard form, which arranges the terms in descending order of their exponents. This format makes it easier to identify the leading term and the degree. For instance, rewriting the polynomial x^2 - 4x^3 + 9x - 12x^4 + 63 in standard form would help quickly identify that the leading term is -12x^4.