Problem 21 - Elementary and Intermediate Algebra

Question

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. See Example \(1 .\) $$ \frac{3}{2} z^{2} $$

Step-by-Step Solution

Answer

The expression \( \frac{3}{2} z^{2} \) is a monomial.

Step 1: Define Polynomial Terms

A monomial is a polynomial with one term. A binomial has two terms, a trinomial has three terms, and anything else doesn't fit these categories.

Step 2: Analyze the Given Expression

Consider the expression \( \frac{3}{2} z^{2} \). It has only one term, which is \( \frac{3}{2} z^{2} \).

Step 3: Classify the Polynomial

Since the expression \( \frac{3}{2} z^{2} \) has only one term, it is classified as a monomial.

Key Concepts

Monomial

A monomial is one of the simplest forms of a polynomial. It consists of a single term, which can be a number, a variable, or the product of a number and one or more variables raised to a power.
Understanding monomials is crucial because they serve as the building blocks for more complex polynomials. Monomials can look like:

• A constant, such as 7.
• A variable, such as \( x \).
• Or an expression like \( 4xy^{2} \).

Each example here includes only one term, regardless of its structure.

Characteristics of a Monomial

A few key features help identify monomials:

• There is no addition or subtraction within the term.
• The variables may have non-negative integer exponents.
• They can be multiplied with each other to form new monomials.

A typical example from the exercise would be \( \frac{3}{2} z^2 \), which is a single term. Therefore, it neatly fits the definition of a monomial.

Binomial

A binomial is a step up in complexity from a monomial. It consists of exactly two terms, usually separated by a plus or minus sign. You can think of a binomial as a small family of terms working together.

Characteristics of a Binomial

• The defining feature is the presence of two distinct terms.
• These terms are separated by either "+" or "-" signs.
• Like monomials, each term can be a constant, a variable, or a combination of numbers and variables.

Examples include:

• \( x + 5 \)
• \( 3y - z \)

In these examples, notice how each expression contains exactly two separate parts — one being added or subtracted from the other. Binomials are important in many algebraic operations, such as factoring and expanding expressions.

Trinomial

A trinomial is a slightly more intricate form of a polynomial than a monomial or a binomial. It consists of three unique terms combined by addition or subtraction. The term "tri" is indicative of its three parts.

Characteristics of a Trinomial

• A trinomial has three separate terms.
• These terms are typically added or subtracted from each other.
• It can include constants, variables, and products of variables with different powers.

For example, a common trinomial might be:

• \( x^2 + 3x + 2 \)

Here, the expression comprises three distinct parts: \( x^2 \), \( 3x \), and \( 2 \), each one adding to the complexity of the polynomial. Trinomials play an essential role in algebra, especially when factoring polynomial expressions or solving quadratic equations.