Problem 7 - Algebra and Trigonometry

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial, then list its terms and state its degree. \(x^{2}-3 x+7\)

Step-by-Step Solution

Answer

The polynomial is a trinomial with terms \(x^2\), \(-3x\), and \(7\), and its degree is 2.

Step 1: Identify the type of polynomial

A polynomial can be classified based on the number of terms it contains. A monomial has one term, a binomial has two terms, and a trinomial has three terms. The given polynomial is \(x^2 - 3x + 7\). Count the number of terms to classify it.

Step 2: List the terms of the polynomial

The terms in the polynomial \(x^2 - 3x + 7\) are separated by '+' or '-' signs. Identify each term in the polynomial for further analysis. The terms are: \(x^2\), \(-3x\), and \(7\).

Step 3: Determine the degree of the polynomial

The degree of a polynomial is the highest power of the variable present in the polynomial. For the polynomial \(x^2 - 3x + 7\), examine the exponents of each term. The terms \(x^2\), \(-3x\), and \(7\) have degrees 2, 1, and 0, respectively, so the polynomial's degree is 2.

Key Concepts

Monomial

A monomial is a polynomial with only one term. It's the simplest type of polynomial. Think of a monomial as a single number, a variable, or a product of numbers and variables raised to a power. For example,

• 5
• -2x
• 3x^2

Each of these examples represents a monomial. Notice that there are no addition or subtraction operations connecting multiple parts. When dealing with monomials, we also pay attention to the degree, which is just the sum of the exponents of the variables.
If there are no variables, like in the number 5, it's degree is 0. The degree will help us when we work with polynomials of more than one term, determining the overall degree in relation to others.

Binomial

A binomial consists of exactly two terms. These are connected by either a '+' or a '-' sign. Binomials are one step up from monomials in terms of complexity. For example, consider:

• x + 1
• 3x - 4

Think of binomials as little pieces of math that bring two different terms together.
Each term has its own degree, but when it comes to the binomial, we only focus on the term with the highest degree to classify the degree of the entire binomial. This is a simple way to see which term carries the most influence in terms of variable power.

Trinomial

A trinomial contains three distinct terms. This is why it is slightly more complex compared to monomials and binomials. In the polynomial example you were given,

• you can see it is a trinomial represented by the expression: \(x^2 - 3x + 7\).

Here, the three terms are separated by the '+' and '-' signs.
Each term again has its own degree, just like the others. When looking at a trinomial, you must check all the terms to find the highest degree term which dictates the trinomial's degree.

Degree of a Polynomial

The degree of a polynomial is essential in understanding its behavior and shape. It is defined as the highest power of the variable in the polynomial's terms. In the trinomial \(x^2 - 3x + 7\), inspect each term:

• \(x^2\) has degree 2
• -3x has degree 1
• 7 has degree 0

Among these, the highest degree term is \(x^2\), making the degree of the polynomial 2.
This tells us that the polynomial will have certain properties, like having no more than two roots and possessing a parabolic shape if graphed. It's a critical identifier in polynomial functions and helps guide how these functions behave across different values.