Problem 1 - 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. \(\begin{array}{ll}{\text { Polynomial }} & \quad{\text { Type Terms Degree }} \\\ {\text { 1. } x^{2}-3 x+7}\end{array}\)

Step-by-Step Solution

Answer

Trinomial, terms: \(x^2\), \(-3x\), \(7\), degree: 2.

Step 1: Identify the Polynomial Type

First, count the number of terms in the polynomial \(x^2 - 3x + 7\). A term is a part of an expression separated by addition or subtraction. Here, the polynomial has three terms: \(x^2\), \(-3x\), and \(7\). Since it has three terms, this is a trinomial.

Step 2: List the Terms of the Polynomial

The polynomial \(x^2 - 3x + 7\) consists of the terms \(x^2\), \(-3x\), and \(7\). These are the individual units that make up the polynomial.

Step 3: Determine the Degree of the Polynomial

The degree of a polynomial is the highest power of the variable present. In \(x^2 - 3x + 7\), the highest exponent of \(x\) is \(2\) (from the term \(x^2\)). Therefore, the degree of this polynomial is 2.

Key Concepts

Understanding Monomials

A monomial is the simplest form of a polynomial. It's an algebraic expression that consists of just one term. This one term can include numbers, variables, and even exponents. However, it cannot include any addition or subtraction signs within it. Here are some examples of monomials:

• 5
• 3x
• -2xy^2

In a monomial, you can find coefficients (the numbers) and the variables raised to whole number powers. Importantly, a monomial represents a single, complete, unbreakable piece of an algebraic expression. Its structure is straightforward because there's no splitting with + or - signs.

Exploring Trinomials

Trinomials are a bit more complex than monomials because they contain exactly three terms. These terms are separated by either addition or subtraction signs in an expression. In the polynomial provided, namely \( x^2 - 3x + 7 \), we have a trinomial. Let's break it down:

• First term: \( x^2 \)
• Second term: \(-3x \)
• Third term: \( 7 \)

These terms combine through operations (addition and subtraction) to form a complete algebraic expression known as a trinomial. Trinomials can often be factored or expanded, depending on the requirement in exercises, making them highly versatile in algebraic manipulations.

Degree of a Polynomial

The degree of a polynomial is a crucial concept in algebra. It indicates the highest power of the variable in the expression. Understanding the degree can help in predicting the behavior of polynomials and their graphs. For the trinomial \( x^2 - 3x + 7 \), the degree is 2 because the highest exponent of the variable \( x \) is found in \( x^2 \).This concept is key in:

• Determining the graph's shape
• Solving polynomial equations
• Understanding polynomial functions' characteristics

Generally, the degree helps classify polynomials. It tells us how many solutions (or roots) the equation might have and gives insights into the polynomial's end behavior in graphs.