Problem 18 - Beginning and Intermediate Algebra

Question

For each polynomial, identify each term in the polynomial, the coefficient and degree of each term, and the degree of the polynomial. $$4 d^{2}+12 d-9$$

Step-by-Step Solution

Answer

The polynomial \(4d^2 + 12d - 9\) has three terms: \(4d^2\), \(12d\), and \(-9\). The coefficients and degrees of each term are as follows: For \(4d^2\), the coefficient is 4 and the degree is 2; for \(12d\), the coefficient is 12 and the degree is 1; for \(-9\), the coefficient is -9 and the degree is 0. The degree of the polynomial is the highest degree of its terms, which in this case is 2.

Step 1: Identify each term in the polynomial

The polynomial \(4d^2 + 12d - 9\) can be split into three terms: \(4d^2\), \(12d\), and \(-9\).

Step 2: Identify the coefficient and degree of each term

The polynomial has 3 terms and for each term: 1. The first term is \(4d^2\). The coefficient is 4, and the degree of this term is 2. 2. The second term is \(12d\). The coefficient is 12, and the degree of this term is 1. 3. The third term is \(-9\). The coefficient is -9, and the degree of this term is 0 (since it is a constant).

Step 3: Identify the degree of the polynomial

The degree of a polynomial is the highest degree of its terms. In this case, the highest degree is 2 (from the term \(4d^2\)). Therefore, the degree of the polynomial is 2.

Key Concepts

Polynomial Terms

In mathematics, a polynomial is essentially a sum of terms. These terms form the basic building blocks of a polynomial expression. Each term in a polynomial consists of constants or variables raised to whole number powers. For example, in the polynomial \(4d^2 + 12d - 9\), we have three distinct terms:

• \(4d^2\)
• \(12d\)
• \(-9\)

Each of these components is referred to as a 'term.' These terms are combined through addition or subtraction, resulting in what we know as a polynomial. It's imperative to recognize and separate these terms when you first encounter a polynomial, as each will have its own specific properties and characteristics. Identifying individual terms is the first step in analyzing any polynomial expression.

Coefficient

The coefficient in a polynomial term is the numerical part that multiplies the variable or variables. Essentially, it tells you how many times that specific term is to be counted. If the term involves a variable, the coefficient is the number placed in front of that variable. Let's explore the coefficients in our example polynomial \(4d^2 + 12d - 9\):

• In the term \(4d^2\), the coefficient is 4.
• In the term \(12d\), the coefficient is 12.
• In the constant term \(-9\), the coefficient is -9.

Even though a constant term does not contain a variable, it can still be thought of as having a coefficient. Specifically, its coefficient is the constant itself. Coefficients are crucial because they affect the magnitude and direction (if they are negative) of the polynomial's terms.

Constant Term

In a polynomial, the constant term is the term without any variables. It's essentially the term made up of just a number. In our example polynomial, \(4d^2 + 12d - 9\), the constant term is \(-9\).

• The constant term is always at the end if the polynomial is arranged in standard form.
• The degree of a constant term is zero because it lacks a variable.

The constant term influences the vertical shift of the graph of the polynomial on a coordinate plane. Since it doesn't change with different values of the variable, its role is static yet essential in the overall polynomial structure.

Degree of a Term

The degree of a term within a polynomial refers to the highest power of the variable in that term. This concept helps in determining the behavior of the polynomial when graphed and offers insight into the polynomial's characteristics.
Let's look at the degrees of each term in our polynomial \(4d^2 + 12d - 9\):

• The term \(4d^2\) has a degree of 2 because the variable \(d\) is raised to the second power.
• The term \(12d\) has a degree of 1, as \(d\) is raised to the first power.
• The constant term \(-9\) has a degree of 0, as there is no variable present.

In a polynomial, knowing the degree of each term assists in finding the overall degree of the polynomial. The overall degree is simply the highest degree of any single term within that polynomial, which, in this case, is 2.