- The degree of a term \(ax^n\) is the exponent \(n\).
- The degree of a polynomial is the highest degree of its terms:
- Degree of \(6x^8 = \boxed{8}\)
- Degree of \(4x^7 = \boxed{7}\)
- Degree of \(1 = \boxed{0}\)
- Degree of \(-x^5 = \boxed{5}\)
- Degree of \(6x^8 + 4x^7 + 1 - x^5 = \boxed{8}\)
Explanation
- Definitions and Analysis Plan
The degree of a monomial (a single term) \(ax^n\) (where \(a
eq 0\)) is simply the exponent \(n\). If the term is a non-zero constant, it can be written as \(ax^0\), meaning its degree is \(0\). The degree of a polynomial is defined as the highest degree among all its individual non-zero terms. We will analyze the terms \(6x^8\), \(4x^7\), \(1\), and \(-x^5\) individually and then determine the degree of the entire polynomial.
- Determining the Degree of Each Term
- For the term \(6x^8\), the exponent of \(x\) is \(8\), so the degree is \(8\).
- For the term \(4x^7\), the exponent of \(x\) is \(7\), so the degree is \(7\).
- For the term \(1\), we can write it as \(1 \cdot x^0\), so the exponent of \(x\) is \(0\), making the degree \(0\).
- For the term \(-1x^5\), the exponent of \(x\) is \(5\), so the degree is \(5\).
- Determining the Degree of the Polynomial
The polynomial is \(P(x) = 6x^8 + 4x^7 - x^5 + 1\). To find the degree of the polynomial, we compare the degrees of its individual terms: \(8, 7, 0, \text{ and } 5\). The highest value among these degrees is \(8\). Therefore, the degree of the polynomial is \(8\).
Examples
In real-world applications, identifying the degree of a polynomial is essential for predicting the long-term behavior of a system. For example, when an engineer models the trajectory of a projectile using a quadratic equation (degree 2), the degree tells them it is a parabola. If an economist uses a polynomial to model market growth, a higher degree polynomial might capture more complex fluctuations over time. Understanding degrees helps in choosing the right mathematical models for different scientific or financial problems.