Problem 10 In the following exercises, dete... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 10

In the following exercises, determine the degree of each polynomial. $$5 c^{3}+11 c^{2}-c-8$$

Short Answer

Expert verified

Step by step solution

Identify Each Term

Identify each term in the polynomial. The polynomial given is $$5 c^{3}+11 c^{2}-c-8$$The terms here are 5c^3, 11c^2, -c, and -8.

Determine the Degree of Each Term

Determine the exponent of the variable in each term, which gives the degree of that term. For this polynomial, the degrees are:- For the term 5c^3: the degree is 3.- For the term 11c^2: the degree is 2.- For the term -c: the degree is 1.- For the term -8: the degree is 0 (since there is no variable, it is a constant term).

Identify the Highest Degree

The degree of the polynomial is the highest degree of any term in the polynomial. From the previous step, we have degrees 3, 2, 1, and 0. The highest degree among these is 3.

State the Degree of the Polynomial

The degree of the polynomial is 3.

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.

Identifying Polynomial Terms

Polynomials are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. To understand a polynomial, you first need to break it down into its individual terms.
Let's look at our polynomial: \[5c^{3} + 11c^{2} - c - 8\].
A term in this context is any part of the polynomial separated by a plus or minus sign.
So, for the polynomial above, the terms are:

$5c^{3}$
$11c^{2}$
$-c$
$-8$

Now, you can see each term consists of a coefficient (the numerical part) and, in some cases, variables raised to a power (called the exponent).

Degree of Polynomial Terms

The degree of a polynomial term is the exponent of the variable in that term. If there is more than one variable in the term, the degrees are summed.
Let's look at each term from our polynomial:

For $5c^{3}$, the degree is 3 because the exponent of $c$ is 3.
For $11c^{2}$, the degree is 2 because the exponent of $c$ is 2.
For $-c$, the degree is 1 because $c$ without an exponent implies $c^{1}$.
For $-8$, the degree is 0 because there is no variable part in the term.

The concept of degree is important because it tells you the highest power of the variable in the polynomial, which in turn gives you an idea of the polynomial鈥檚 complexity and behavior when plotted.

Highest Degree in a Polynomial

To determine the degree of a polynomial, we look for the highest degree among its individual terms.
From our polynomial \[5c^{3} + 11c^{2} - c - 8\],
we see that the terms have degrees 3, 2, 1, and 0 respectively.
The highest degree is clearly 3.
This means the degree of the polynomial \[5c^{3} + 11c^{2} - c - 8\] is 3.
Knowing the highest degree of the polynomial helps in understanding its leading behavior.
For example, in a polynomial of degree 3, the function will generally grow proportionally with a cubic factor as the variable increases. This degree also tells you the number of roots (or solutions) the polynomial can have, which is up to the polynomial's degree.

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

In the following exercises, determine the degree of each polynomial. $$5 c^{3}+11 c^{2}-c-8$$

Short Answer

Step by step solution

Identify Each Term

Determine the Degree of Each Term

Identify the Highest Degree

State the Degree of the Polynomial

Key Concepts

Identifying Polynomial Terms

Degree of Polynomial Terms

Highest Degree in a Polynomial

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Statistics

Pure Maths

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.