Problem 32 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The polynomial \(a^3b - ab^3\) is a binomial.

Step 1: Understanding Polynomial Terms

The given expression is \(a^3b - ab^3\). First, identify the separate terms. Here, we have two terms: \(a^3b\) and \(-ab^3\).

Step 2: Counting the Terms

A polynomial's classification depends on the number of terms. Count the terms in the expression: \(a^3b - ab^3\) has two terms.

Step 3: Classification by Number of Terms

- A monomial has one term. - A binomial has two terms. - A trinomial has three terms. - If it doesn't fit any of these, it is none of these. Since the expression has two terms, it is a binomial.

Key Concepts

Monomials

Monomials are the simplest type of polynomials. A monomial is essentially a single term that consists of a constant, a variable, or the product of constants and variables.
Here are the main features of a monomial:

• Just one non-zero term
• No addition or subtraction within the term
• Variables in the term can be raised to a whole number exponent

Examples of monomials include \(5x^3\), \(7\), and \(-2yz\).
Notice that monomials can also include multiple variables or higher powers as long as they stand alone as a single term.
For instance, \(3xyz^2\) is still a monomial because it's one term with several variables interacted through multiplication and exponentiation.
It's important to remember that monomials play a foundational role in algebra, and understanding them prepares you for tackling more complex polynomial expressions.

Binomials

Binomials are a step up from monomials, as they contain exactly two distinct terms.
These terms are usually linked together by a plus or minus sign.

• Consists of exactly two terms
• Addition or subtraction separates the terms
• Each term can include constants, variables, and exponents

An example of a binomial is: \(2x^2 + 3y\) or \(-4a + b^3\).
What differentiates binomials from other polynomials is this fixed two-term structure. Binomials are widely used in algebra to encounter problems, especially when it comes to factorization and expansion using formulas like \(a^2 - b^2 = (a-b)(a+b)\).
With practice, identifying and working with binomials can become second nature, simplifying many algebraic problems.

Trinomials

Trinomials add another layer, consisting of three separate terms within the polynomial.
The distinguishing factor with trinomials is that they often appear in quadratic equations and take forms suitable for factoring.

• Made up of exactly three terms
• Connected by addition or subtraction
• Each term can still contain variables and constants

An academic example of a trinomial is \(x^2 + 5x + 6\).
Trinomials are crucial in the study of algebra because they often represent equations that can be graphed as parabolas. Recognizing a trinomial quickly allows you to apply various algebraic methods such as factoring or using the quadratic formula to solve related equations.
As you become more adept at identifying and manipulating trinomials, you'll unlock solutions to many classic algebra problems.