Problem 10 - Precalculus Mathematics for Calculus

Question

Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree. $$\frac{1}{2} x^{7}$$

Step-by-Step Solution

Answer

Monomial; term: \( \frac{1}{2} x^{7} \); degree: 7.

Step 1: Identify the Type of Polynomial

First, we need to determine the number of terms in the polynomial. The expression \( \frac{1}{2} x^{7} \) consists of a single term. Thus, it is classified as a monomial.

Step 2: List the Terms

The given polynomial consists only of one term. The term of the polynomial is \( \frac{1}{2} x^{7} \).

Step 3: Determine the Degree

The degree of a polynomial is determined by the highest power of the variable in the expression. In the polynomial \( \frac{1}{2} x^{7} \), the variable \( x \) is raised to the power of 7, making the degree 7.

Key Concepts

Monomial

A monomial is a type of polynomial with only one term. It's the simplest form of a polynomial. Monomials can include numbers, variables, or a combination of both.
Each part of the term is important to understand:

• **Number**: This part is called the coefficient. It can be any real number, including fractions like in \( \frac{1}{2} x^7 \).
• **Variable**: This is the unknown value that can change, often represented by letters like \( x \). Variables can have exponents (powers) in monomials.
• **Exponent**: This is the power to which the variable is raised. It tells how many times the variable is multiplied by itself. In the case of \( \frac{1}{2} x^7 \), the exponent is 7.

Monomials are important because they serve as the building blocks for more complex polynomials such as binomials and trinomials.

Degree of Polynomial

The degree of a polynomial is a key concept that helps describe and compare different polynomials. It is defined as the highest power of the variable in a polynomial expression. For the monomial \( \frac{1}{2} x^7 \), the degree is straightforward to determine:

• The exponent of the variable \( x \) is 7, which makes the degree 7.

Understanding the degree is crucial because it can indicate:

• **Behavior of the polynomial**: The degree can tell us how many roots (zeroes) a polynomial can have and how the graph of the polynomial might look (like how many turns it may take).
• **Simple classification**: With polynomials, higher degrees generally mean more complexity.

This concept plays a fundamental role in algebra and calculus, helping you solve equations more effectively.

Terms of a Polynomial

A polynomial is made up of terms added or subtracted together. Each term is a combination of:

• a coefficient,
• a variable,
• and an exponent.

In the exercise example, \( \frac{1}{2} x^7 \) is a single term, which classifies the expression as a monomial.
Understanding the 'terms' allows you to:

• **Identify the type of polynomial**: By counting the number of terms, you can classify a polynomial. One term is a monomial, two terms a binomial, and three terms a trinomial.
• **Simplify and solve equations**: Knowing how to combine and simplify terms is fundamental in solving mathematical problems.

Terms can be combined with others that share the same variable and exponent, helping to simplify expressions and equations.