Problem 12 Determine whether each is a mono... [FREE SOLUTION]

Chapter 6: Problem 12

Determine whether each is a monomial, a binomial, or a trinomial. $$v^{4}+7 v^{2}+6$$

Short Answer

Expert verified

The given expression $v^4 + 7v^2 + 6$ has three terms, so it is a trinomial.

Step by step solution

Count the number of terms in the expression

To count the number of terms, we can look at the given expression and identify the different expressions separated by '+' and '-' signs. $v^4 + 7v^2 + 6$ has three terms: $v^4$, $7v^2$, and $6$.

Classify the expression

Since the expression has three terms, it is classified as a trinomial. So, $v^4 + 7v^2 + 6$ is a trinomial.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Monomials

A monomial is a polynomial with exactly one term. This means there are no addition or subtraction signs separating different expressions. Monomials can involve:

Constants, like 5 or -3
Variables, such as x or y
Products of constants and variables, like 4x or -7y^3

To be a valid monomial, the exponent of any variable present must be a non-negative integer. This means that terms like $4x^{-2}$ or $5\sqrt{x}$ are not considered monomials because they involve negative or fractional exponents. Understanding monomials is fundamental, as they're the building blocks for more complex polynomial expressions.

Binomials

A binomial is a polynomial expression containing exactly two terms. These terms are often separated by either a plus (+) or a minus (-) sign. For example, $3x + 4$ and $7a^2 - 5b$ are both binomials. Binomials can represent simple mathematical ideas but are also incredibly useful in more complex algebraic manipulations. They appear frequently in operations such as:

Factoring quadratic equations
Expanding expressions using the distributive property
Applying the binomial theorem

The key to recognizing a binomial is counting the terms carefully: there must be exactly two standalone segments in the expression. Each term in a binomial might be a monomial itself.

Trinomials

A trinomial is a specific type of polynomial that includes exactly three terms. Like with our given example $v^4 + 7v^2 + 6$, each term in a trinomial is separated by a plus (+) or minus (-) sign. This type of polynomial is frequently encountered in algebra, especially in quadratic expressions. Common applications of trinomials include:

Solving quadratic equations using factoring
Completing the square
Graphing parabolas

When working with trinomials, understanding how to identify and separate each term is crucial for further algebraic processes such as factoring or simplifying expressions. The term's exponents and coefficients play a significant role in determining the behavior and characteristics of the trinomial.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Determine whether each is a monomial, a binomial, or a trinomial. $$v^{4}+7 v^{2}+6$$

Short Answer

Step by step solution

Count the number of terms in the expression

Classify the expression

Key Concepts

Monomials

Binomials

Trinomials

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Applied Mathematics

Logic and Functions

Pure Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.