Problem 7 - College Algebra

Question

For the following exercises, identify the degree of the polynomial. $$ -625 a^{8}+16 b^{4} $$

Step-by-Step Solution

Answer

The degree of the polynomial is 8.

Step 1: Identify degree of each term

A polynomial is made up of terms like \[ -625a^8 \] and \[ 16b^4 \].For each term, we determine the degree by looking at the exponents of the variables. The degree of \[ -625a^8 \] is \(8\) since the variable \(a\) is raised to the power \(8\). Similarly, the degree of \[ 16b^4 \] is \(4\) because \(b\) is raised to the power \(4\).

Step 2: Determine the largest degree

The degree of a polynomial is defined by the highest degree of its terms. In the polynomial \[ -625a^8 + 16b^4 \] the degrees of the terms are \(8\) and \(4\). The highest of these degrees is \(8\).

Step 3: State the degree of the polynomial

Based on Step 2, where we determined that the highest degree among the terms is \(8\), the degree of the polynomial \[ -625a^8 + 16b^4 \] is \(8\).

Key Concepts

Understanding Polynomial Terms

A polynomial is a mathematical expression composed of one or more terms. These terms are individual parts of the polynomial, each consisting of a constant coefficient and variable(s) raised to a power. In simpler terms, each term can be seen as a "building block" of the polynomial.

Consider the polynomial given in the exercise: \(-625a^8 + 16b^4\).

• \(-625a^8\) is one term.
• \(16b^4\) is another term.

Each term contains a variable and a coefficient:

• For \(-625a^8\), the coefficient is \(-625\) and the variable part is \(a^8\).
• For \(16b^4\), the coefficient is \(16\) and the variable part is \(b^4\).

Identifying terms is an essential step as it allows us to analyze and determine other properties of the polynomial, such as its degree.

Defining the Highest Degree

The degree of a polynomial is a crucial concept when understanding polynomials. To find it, you look at the highest degree among all the terms of the polynomial.

Each term has its own degree, based on the exponents of the variables it contains. The term with the largest sum of exponents determines the degree of the whole polynomial.

For example, in the polynomial \(-625a^8 + 16b^4\), you determine the degree of each term:

• The degree of \(-625a^8\) is \(8\) because the variable \(a\) has an exponent of \(8\).
• The degree of \(16b^4\) is \(4\) because the variable \(b\) has an exponent of \(4\).

The term with the highest degree in this polynomial is \(-625a^8\) with a degree of \(8\). Therefore, the polynomial itself also has a degree of \(8\).

Recognizing the highest degree in a polynomial is crucial as it provides information about the polynomial's behavior and its graphical representation.

Exploring Exponents of Variables

Exponents in polynomials indicate how many times a variable is multiplied by itself. They are central to determining the degree of a term and the entire polynomial.

Look at individual terms in a polynomial, like \(-625a^8\) and \(16b^4\). Here, the exponents are:

• For \(a^8\), the exponent is \(8\), meaning \(a\) is utilized as a factor eight times.
• For \(b^4\), the exponent is \(4\), meaning \(b\) is utilized as a factor four times.

By understanding exponents, we can quickly assess and compare the degrees of different terms within a polynomial.

In summary, the exponents of variables not only define the degree of individual terms but also play a decisive role in determining the overall degree of the polynomial.