Problem 7 - Algebra for College Students

Question

Determine the degree of the given polynomials. $$8 x^{6}+9$$

Step-by-Step Solution

Answer

The degree of the polynomial is 6.

Step 1: Identify the Terms of the Polynomial

The given polynomial is \(8x^6 + 9\). It consists of two terms: \(8x^6\) and \(9\).

Step 2: Determine the Degree of Each Term

The degree of the term \(8x^6\) is 6, as the exponent of \(x\) is 6. The degree of the constant term \(9\) is 0 since it can be written as \(9x^0\).

Step 3: Identify the Degree of the Polynomial

The degree of a polynomial is determined by the term with the highest degree. In this polynomial, the term with the highest degree is \(8x^6\), which has a degree of 6.

Key Concepts

Understanding Polynomial Terms

In the context of algebra, understanding what makes up a polynomial is crucial. A polynomial consists of terms. Terms are the individual parts of a polynomial that are added or subtracted together.
For example, in the polynomial \(8x^6 + 9\), there are two distinct terms: \(8x^6\) and \(9\).

• The term \(8x^6\) includes a coefficient (8), a variable \(x\), and an exponent (6).
• The term \(9\) is a constant term; it doesn't have a variable associated with it.

Identifying the terms is the first step to understanding a polynomial. Each term can significantly impact the characteristics of the polynomial, such as its degree or how it behaves graphically.In polynomials, it's important to separate terms correctly. This will allow you to analyze each one individually for further concepts like determining exponents and degrees.

Exponents in Polynomials

Exponents play a significant role in defining the behaviour of polynomials. The exponent in a polynomial term indicates the power to which the variable (usually \(x\)) is raised. For example, in the term \(8x^6\), the exponent is 6.

• The higher the exponent, the more our polynomial's graph may change direction or curvature.
• When a term doesn't visibly show an exponent, it is assumed to have an exponent of 1, hence \(x^1\).

In polynomials, the exponent is always a non-negative integer. Negative or fractional exponents would disqualify the expression from being a polynomial.
This means when analyzing a polynomial, each variable present must have a whole number indicating its exponent.Understanding the role of exponents helps in determining not only the degree of terms but also the overall degree of the polynomial. The term with the highest exponent controls many aspects of the polynomial's properties.

Constant Term Degree Explanation

The constant term in a polynomial is unique because it lacks a variable. While variable terms have exponents, the constant term includes an implied exponent of 0, represented as \(x^0\). For instance, in the polynomial \(8x^6 + 9\), the constant term is \(9\).

• The degree of any given constant term is always 0.
• This is because multiplying by \(x^0\) is equivalent to multiplying by 1, leaving the constant unchanged.

Often, students might mistake constant terms as independent of degrees because they have no visible variables, but recognizing them as zero-degree terms is crucial for correctly determining the degree of the entire polynomial.While constant terms don't influence the highest degree of the polynomial, understanding their zero-degree status can help clearly form the polynomial's complete degree structure, complementing the understanding of exponents in variable terms.