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Chapter 6: Problem 7

Determine whether each is a monomial, a binomial, or a trinomial. $$3 x-7$$

Short Answer

Expert verified
The given expression \(3x - 7\) has two terms, \(3x\) and \(-7\). Therefore, it is a binomial.

Step by step solution

01

Identify the expression

The given expression is \(3x - 7\).
02

Count the number of terms

In the expression \(3x - 7\), there are two terms: \(3x\) and \(-7\). These terms are separated by a minus (-) sign.
03

Classify the expression

Since there are two terms in the expression, it is a binomial. So, the given expression, \(3x - 7\), is a binomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomial
A monomial is the simplest form of a polynomial and is a single term expression. It consists of a product of numbers and variables with non-negative integer exponents. To identify a monomial, check if the expression has just one term without addition or subtraction. Here are some distinctive features of monomials:
  • They do not contain addition or subtraction expressions.
  • They can include constants (numbers), variables, or both together, such as 5, x, or 7xy2.
  • The variables in a monomial can have exponents, as long as they are whole numbers.
For example, the terms: 6, -3a, and 4b3c are all considered monomials. Understanding monomials is crucial as they form the building blocks for more complex polynomials like binomials and trinomials.
Binomial
A binomial is a type of polynomial that includes exactly two terms, separated by either a plus (+) or minus (-) sign. The expression can include numbers, variables, or both. Recognizing a binomial is pretty straightforward. You simply count two separate parts that are added or subtracted.
  • They always consist of two monomials together.
  • Each term can have different variables or powers.
  • Examples include expressions like a + b or 5x2 - 3.
In the original exercise, we saw the expression 3x - 7. By counting the two distinct terms, 3x and -7, it is clear that it's a binomial because of the presence of just two terms. Binomials are commonly used in various algebraic operations like factoring and expanding expressions.
Trinomial
A trinomial is a polynomial that has three distinct terms. Like binomials, the terms in a trinomial are also separated by addition or subtraction signs. Identifying a trinomial involves counting the separate components in the expression.
  • It consists of exactly three monomials.
  • The terms can appear in any combination of constants and variables.
  • Examples of trinomials include x2 + 3x + 2 and 5a + 4b - 1.
It's important to distinguish trinomials as their own category because they often appear in algebraic problems, especially those involving quadratic equations and factoring techniques. Understanding how to identify and work with trinomials is key to mastering polynomial expressions in algebra.

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