Problem 626 For the polynomial \(8 y^{4}-3 y... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 626

For the polynomial $8 y^{4}-3 y^{2}+1$ (a) Is it a monomial, binomial, or trinomial? (b) What is its degree?

Short Answer

Expert verified

(a) Trinomial, (b) Degree 4.

Step by step solution

Identify the Number of Terms

The polynomial given is $8 y^{4}-3 y^{2}+1$. Determine how many terms are in this polynomial. For this polynomial, the terms are $8 y^{4}$, $-3 y^{2}$, and $1$. Thus, the polynomial has three terms.

Classify the Polynomial

Based on the number of terms identified, classify the polynomial. A monomial has one term, a binomial has two terms, and a trinomial has three terms. Since there are three terms, this polynomial is a trinomial.

Identify the Degree of Each Term

The degree of a term is the exponent of the variable in that term. For $8 y^{4}$, the degree is 4. For $-3 y^{2}$, the degree is 2. For the constant term $1$, the degree is 0.

Determine the Degree of the Polynomial

The degree of a polynomial is the highest exponent of the variable in the polynomial. From the identified degrees, the highest is 4. Hence, the polynomial has a degree of 4.

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.

Trinomial Definition

A trinomial is a special type of polynomial. A polynomial is an expression that consists of variables, coefficients, and non-negative integer exponents. They can be made up of multiple terms combined using addition or subtraction. The term 'trinomial' specifically describes a polynomial that contains exactly three terms.

For instance, in the polynomial given in the exercise, $8y^4 - 3y^{2} + 1$, we can see that it contains three distinct terms: $8y^4$, $-3y^2$, and $1$.

The presence of exactly three terms classifies the polynomial as a trinomial. This is why, in step 2 of the solution, the polynomial was classified as a trinomial.

Degree of Polynomial

When we talk about the degree of a polynomial, we are referring to the highest exponent of the variable in any of its terms. This tells us a lot about the polynomial鈥檚 behavior, especially for large values of the variable.

Each term in a polynomial can have its power or degree. For example, in the term $8y^4$, the exponent is 4, so the degree of this term is 4.

Let's look at the terms in our example polynomial: $8y^4 - 3y^2 + 1$:

$8y^4$: Degree 4
$-3y^2$: Degree 2
$1$ (constant term): Degree 0

The highest degree among these terms is 4, which means the degree of the polynomial is 4.

Understanding the degree of a polynomial helps in many areas, including graphing and solving polynomial equations.

Terms in Polynomials

A polynomial is composed of several terms. Each term is a product of a constant coefficient and a variable raised to a non-negative integer power. For example, in the polynomial $8y^4 - 3y^{2} + 1$:

$8y^4$ is a term with coefficient 8 and variable $y$ raised to the power 4
$-3y^2$ is a term with coefficient -3 and variable $y$ raised to the power 2
$1$ is a constant term with a coefficient of 1 and a variable power of 0

Identifying individual terms in a polynomial is crucial because each term's degree and coefficient determine the polynomial's overall structure and properties.

Moreover, the nature of the terms (whether they include variables or are constants) plays a major role in the polynomial classification and degree analysis.

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

For the polynomial \(8 y^{4}-3 y^{2}+1\) (a) Is it a monomial, binomial, or trinomial? (b) What is its degree?

Short Answer

Step by step solution

Identify the Number of Terms

Classify the Polynomial

Identify the Degree of Each Term

Determine the Degree of the Polynomial

Key Concepts

Trinomial Definition

Degree of Polynomial

Terms in Polynomials

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Decision Maths

Logic and Functions

Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.