Problem 18 - Elementary and Intermediate Algebra

Question

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. See Example \(1 .\) $$ 3 y-5 $$

Step-by-Step Solution

Answer

Binomial

Step 1: Identify the number of terms

Count the number of terms in the polynomial. A term in a polynomial is any coefficient or constant with a variable. In the expression \(3y - 5\), there are two terms: \(3y\) and \(-5\).

Step 2: Classify based on number of terms

Determine the classification based on the number of terms identified in Step 1. A monomial has one term; a binomial has two terms; a trinomial has three terms; and any polynomial with more than three terms is classified based on how many terms it has, usually as 'polynomial'. Since \(3y - 5\) has two terms, it is classified as a binomial.

Key Concepts

Monomial

A monomial is the simplest type of polynomial. It consists of only one term. This means that a monomial could be a single number, a variable, or the product of a number and one or more variables.
For example, consider the following expressions as monomials:

• 5
• x
• 7y^2

Because monomials have only one term, they cannot include addition or subtraction.
An easy way to remember this is that a monomial is 'mono', or single, implying that there is just one element involved in the expression.

Binomial

A binomial is a type of polynomial that has exactly two terms. This term also stems from the word 'bi', which signifies two. Binomials can be thought of as the bread and butter of polynomials; they are often involved in algebraic operations due to their simplicity but varied application possibilities.
For example, consider the expression \(3y - 5\). Here, we can identify the two terms: \(3y\) and \(-5\). Because it has two terms, this expression is classified as a binomial.
To determine if an expression is a binomial, count the separate parts that are either added or subtracted.
Remember, it's all about the number of terms, not what's inside of them.

Trinomial

A trinomial, as the name suggests, is a polynomial with three terms. The 'tri' prefix is your hint to remember this classification. Trinomials are important in mathematical factorization and can be seen in quadratic expressions.
Take, for example, the expression \(x^2 + 4x + 4\). This has three terms: \(x^2\), \(4x\), and \(4\). Together, they form a trinomial.
When identifying trinomials, ensure that you can distinctly count three separate terms in the expression.

• These terms can be coefficients, variables, or constants.
• Always look for '+' or '-' signs separating different terms.

This classification helps significantly in solving and simplifying more complex algebraic expressions.