Problem 2 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. What is the degree of the polynomial \(5 x^{4}-2 x-7 ?\).

Step-by-Step Solution

Answer

The degree of the polynomial \(5x^{4}-2x-7\) is 4.

Step 1 - Identify the terms in the polynomial

The polynomial is \(5x^{4}-2x-7\). Break it down into its terms. The three terms in the polynomial are \(5x^{4}\), \(-2x\), and \(-7\).

Step 2 - Identify the degree of each term

The degree of a term is the power of the variable in the term. For the term \(5x^{4}\), the degree is 4. For the term \(-2x\), the degree is 1. For the constant term \(-7\), the degree is implicitly 0, because there's no variable associated with it. So, the degrees of the terms are 4, 1, and 0 respectively.

Step 3 - Recognize the highest degree

The degree of the polynomial is the highest degree of its individual terms. Comparing the degrees of the terms we identified in the previous step, we see that the first term has the highest degree, which is 4.

Key Concepts

Polynomial Terms

When dealing with polynomials, it's crucial to understand the structure by identifying something called polynomial terms. A polynomial is essentially an expression that consists of variables and coefficients, typically arranged in terms of powers. In the polynomial given in our original exercise, which is \(5x^{4}-2x-7\), we identify three distinct terms. These terms are:

• \(5x^{4}\)
• \(-2x\)
• \(-7\)

Each term within a polynomial is separated by a plus \( (+) \) or minus \( (-) \) sign.
It's important to recognize each term because they individually contribute to finding other critical features, such as the degree of the polynomial or the identification of constant terms. Think of terms as the building blocks of a polynomial, each holding its unique characteristics defined by its coefficient and its variable’s power.

Degree of a Term

The degree of a term in a polynomial is determined by the power, or exponent, of the variable within that term. This concept is straightforward: the higher the exponent, the higher the degree of that term. Let's take a look at the terms from the polynomial in our exercise:

• \(5x^{4}\) - The term has a degree of 4 because the variable \(x\) is raised to the power of 4.
• \(-2x\) - This term has a degree of 1 since \(x\) is implicitly \(x^1\).
• \(-7\) - Interestingly, this term does not have a variable, and we identify it as having a degree of 0.

The degree gives us insight into how the term will behave as the value of the variable changes, crucially influencing the polynomial's shape graphically. In summary, to find the degree of a term, simply look at the exponent of its variable(s).

Constant Term Degree

A constant term is quite a unique member of polynomial terms. This is because it does not contain a variable, meaning its degree is always zero. In the context of the polynomial \(5x^{4}-2x-7\), the constant term is \(-7\).
This implies:

• The degree of \(-7\) is 0, as it lacks a variable or power.

The concept of constant term degree is essential because it highlights that every term in a polynomial contributes to the expression in a different way.
While constant terms do not affect the overall degree of a polynomial since they have no variable part, they are vital when considering the entire structure and evaluation of a polynomial expression. Understanding this makes solving polynomial equations and manipulating them much more straightforward.