Problem 31 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The polynomial is a trinomial.

Step 1: Identify the Number of Terms

A polynomial is classified based on the number of terms it contains. We will count the number of terms in the polynomial \(2a^{2} - 3ab + b^{2}\). A term is a product of numbers and variables, and in this polynomial, the terms are \(2a^{2}\), \(-3ab\), and \(b^{2}\). This gives us a total of three terms.

Step 2: Classify Based on the Number of Terms

Now that we have identified the number of terms, we can classify the polynomial as follows:- A **monomial** has exactly one term.- A **binomial** has exactly two terms.- A **trinomial** has exactly three terms.Since our polynomial \(2a^{2} - 3ab + b^{2}\) has three terms, it is classified as a trinomial.

Key Concepts

monomial

A monomial in algebra is a polynomial with exactly one term. Think of it as a single building block. This term can be a number, a variable, or a product of numbers and variables. Some examples include:

• \( 3x \)
• \( -7 \)
• \( 5xy^2 \)

What makes monomials unique is their simplicity.
They don't involve addition or subtraction with other terms. Instead, everything is just multiplied or sometimes "stacked" with variables and exponents.
In essence, monomials are the simplest type of polynomial.

binomial

A binomial consists of exactly two terms. The prefix 'bi' suggests this definition, similar to words like 'bicycle'. In a binomial, each term can be a constant, a variable, or a combination of both. These terms are usually connected by either a plus (+) or minus (-) sign. Here are examples of binomials:

• \( x + 5 \)
• \( 3a - 2b \)
• \( y^2 - y \)

Binomials are very flexible in algebra.
They can often be seen in expressions or equations where two different quantities are compared or combined.
Recognizing binomials helps understand more complex mathematical expressions.

trinomial

A trinomial is a polynomial that consists of exactly three terms. The prefix 'tri' highlights this as it relates to the number three, much like a 'tricycle'. In trinomials, you will find three terms that can combine variables and constants linked together by addition or subtraction. Examples include:

• \( 2x^2 + 3x + 1 \)
• \( a^2 - ab + b^2 \)
• \( x^3 + 2x + 3 \)

Trinomials are common in algebraic equations, especially quadratic equations.
They often appear in polynomial operations, such as addition, subtraction, or factoring.
Understanding trinomials is crucial for solving complex mathematical problems. By grasping the concept of trinomials, you can navigate algebraic expressions and equations with more confidence.