Problem 11 - Elementary and Intermediate Algebra

Question

Make a term-coefficient-degree table like that shown in Example 1 for each polynomial. $$ 8 x^{2}+x-7 $$ (TABLE NOT COPY)

Step-by-Step Solution

Answer

Create a table with terms \(8x^2\), \(x\), \(-7\); coefficients 8, 1, -7; and degrees 2, 1, 0 respectively.

Step 1: Identify Each Term in the Polynomial

First, we need to identify each term in the polynomial \(8x^2 + x - 7\). The polynomial has three terms: \(8x^2\), \(x\), and \(-7\).

Step 2: Record Coefficients of Each Term

Next, we determine the coefficient of each term. In \(8x^2\), the coefficient is 8. In \(x\), the coefficient is implicitly 1. In \(-7\), the constant term has a coefficient of -7.

Step 3: Determine the Degree of Each Term

Now, identify the degree of each term. The degree of \(8x^2\) is 2 (highest power of \(x\)), the degree of \(x\) is 1, and \(-7\) is a constant term with a degree of 0.

Step 4: Create the Term-Coefficient-Degree Table

Using the information from the previous steps, create a table:| Term | Coefficient | Degree ||---------|-------------|--------|| \(8x^2\) | 8 | 2 || \(x\) | 1 | 1 || \(-7\) | -7 | 0 |This table organizes each term with its respective coefficient and degree.

Key Concepts

Term Identification in Polynomials

Polynomials are algebraic expressions that consist of several terms. To effectively work with polynomials, the first step is to identify each term. A term is a part of the polynomial that can be as simple as a constant number or as complex as a product of numbers and variables raised to a power.
When we look at the polynomial expression \(8x^2 + x - 7\), we can breakdown each component:

• \(8x^2\)
• \(x\)
• \(-7\)

Each of these represents a separate term in the polynomial. Spotting these helps in managing and organizing the information within a polynomial efficiently.

Understanding Coefficients

Coefficients are the numerical factors in terms of a polynomial. They multiply the variable or variables in the term.
In the polynomial \(8x^2 + x - 7\), each term has a specific coefficient:

• In \(8x^2\), the coefficient is 8.
• In \(x\), even though the number isn't written, the coefficient is 1. This is because \(x\) is the same as \(1x\).
• In the term \(-7\), the whole term is a constant, so it’s technically its own coefficient, which is -7.

Understanding coefficients helps in scaling terms, simplifying equations, and solving for variables effectively.

Determining the Degree of a Term

The degree of a term is the exponent of the variable within that term. For polynomials, the degree is an important indicator of the polynomial’s behavior and complexity.
Looking again at the polynomial \(8x^2 + x - 7\), the degree of each term is as follows:

• For \(8x^2\), the degree is 2, since the variable is raised to the power of 2.
• For \(x\), the degree is 1 because it can be rewritten as \(x^1\).
• For the constant term \(-7\), the degree is 0, because there is no variable present (or the variable could be considered as \(x^0\), which equals 1).

Recognizing the degree of each term helps predict how polynomials behave when graphed or during operations like differentiation and integration.

Creating a Polynomial Table

A polynomial table effectively organizes information about the terms in a polynomial, including their coefficients and degrees. This structure simplifies comparisons and calculations by displaying data in one glance.
When creating a polynomial table for \(8x^2 + x - 7\), insert the terms, their coefficients, and their degrees into a structured list or spreadsheet, like so:

• Term: \(8x^2\), Coefficient: 8, Degree: 2
• Term: \(x\), Coefficient: 1, Degree: 1
• Term: \(-7\), Coefficient: -7, Degree: 0

This method makes it easier to compare and contrast the components of complex polynomials and can hugely assist in solving algebraic problems by maintaining a clear overview.