Problem 8 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$5 y^{6}+y^{4}-2 y^{2}-8$$

Step-by-Step Solution

Answer

The degree of the polynomial is 6.

Step 1: Identify the terms in the polynomial

Look at each term in the polynomial separately. The given polynomial is \(5y^6 + y^4 - 2y^2 - 8\). It has four terms: \(5y^6\), \(y^4\), \(-2y^2\), and \(-8\).

Step 2: Determine the degree of each term

The degree of a term in a polynomial is the exponent of the variable in that term. Analyze the terms: \(5y^6\) has a degree of 6, \(y^4\) has a degree of 4, \(-2y^2\) has a degree of 2, and \(-8\) (a constant term) has a degree of 0.

Step 3: Identify the highest degree

To determine the degree of the entire polynomial, select the highest degree from among all the terms. Among \(6\), \(4\), \(2\), and \(0\), the highest degree is \(6\).

Step 4: State the polynomial's degree

Since the highest degree among the terms is 6, the degree of the polynomial \(5y^6 + y^4 - 2y^2 - 8\) is 6.

Key Concepts

Polynomial terms

When assessing a polynomial, the first step is to identify the individual terms that make it up. Each term in a polynomial consists of a numerical coefficient and one or more variables raised to a power, or exponent. In the polynomial given, \(5y^6 + y^4 - 2y^2 - 8\), there are four terms:

• \(5y^6\)
• \(y^4\)
• \(-2y^2\)
• \(-8\)

By breaking down the polynomial in this way, you can clearly see the structure, which helps in understanding the role each component plays.

Constant term

A crucial aspect of polynomials is the constant term. This is the term in the polynomial that does not contain any variables. In our example, the term \(-8\) is the constant term. It's important to remember:

• A constant term is an actual number with no variable part.
• Its role is significant because it represents a fixed value added or subtracted from the rest of the polynomial expression.

In terms of degree, a constant term is considered to have a degree of 0 because there are no variables present to contribute any power.

Variable exponents

Variable exponents are the superscript numbers attached to the variables in polynomial terms, indicating the power to which the variable is raised. They play a key role in determining the degree of each term.

• The term \(5y^6\) has an exponent of 6, indicating variable \(y\) is raised to the sixth power.
• The term \(y^4\) means \(y\) is raised to the fourth power.
• In \(-2y^2\), \(y\) is squared, or raised to the power of 2.

Understanding variable exponents helps categorize each term's degree. The exponents dictate the growth rate of each term as their variables increase in value.

Identifying highest degree

The degree of a polynomial is one of its most defining features. It is determined by identifying the term with the highest exponent of the variable.

• Among the terms in the polynomial \(5y^6 + y^4 - 2y^2 - 8\), \(5y^6\) has the highest degree.
• The highest degree here is 6, making the entire polynomial's degree 6.

Recognizing the highest degree term is essential in analyzing the polynomial’s overarching behavior, such as its end behavior and potential number of roots. This step helps in understanding the complexity and general behavior of polynomial functions.