Problem 15 For each polynomial, determine t... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 5: Problem 15

For each polynomial, determine the mumber of terms and name the coefficients of the terms. $$ x+8 x^{2}+5 x^{3} $$

Short Answer

Expert verified

The polynomial has 3 terms. The coefficients are: 1, 8, and 5.

Step by step solution

Identify the Terms

Examine the given polynomial: $ x + 8x^2 + 5x^3 $. Each distinct addition or subtraction part of this polynomial is considered a term.

Count the Terms

Count the number of terms in the polynomial: $ x $, $ 8x^2 $, and $ 5x^3 $. Thus, the polynomial has 3 terms.

Identify Coefficients

Identify the coefficient of each term. The coefficient is the numerical factor: - For $ x $, the coefficient is 1,- For $ 8x^2 $, the coefficient is 8,- For $ 5x^3 $, the coefficient is 5.

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.

terms in a polynomial

A polynomial is an algebraic expression consisting of variables, coefficients, and exponents. Polynomials are made up of multiple parts called 'terms'.

Each term is separated by a plus (+) or minus (-) sign. For example, consider the polynomial from the exercise: $ x + 8x^2 + 5x^3 $. Here, the terms are:

$ x $
$ 8x^2 $
$ 5x^3 $

The number of terms in a polynomial is simply the count of these separate parts. In the given example, the polynomial has three terms, as identified in the solution.

coefficients

In a polynomial, coefficients are the numerical factors that multiply the variables. They play a crucial role as they signify the quantity of the variable present in each term.

Let's break down the coefficients in the polynomial $ x + 8x^2 + 5x^3 $:

For the term $ x $, the coefficient is 1 (since $ x $ is the same as $ 1x $).
For the term $ 8x^2 $, the coefficient is 8.
For the term $ 5x^3 $, the coefficient is 5.

Identifying the coefficients helps in understanding and solving polynomial equations.

polynomial structure

A polynomial's structure is defined by its terms, degrees, and coefficients. Understanding these elements helps in recognizing the polynomial's form and behavior.

The general structure of a polynomial can be written as:

$ a_n x^n + a_{n-1} x^{n-1} + \ ... + a_1 x + a_0 $, where
- $a_n, a_{n-1}, ... , a_0 $ are coefficients
- $n, n-1, ... , 1 $ are the exponents (or degrees)

For our example $ x + 8x^2 + 5x^3 $, the structure is identified as:

$ x $ - a term with degree 1 and coefficient 1.
$ 8x^2 $ - a term with degree 2 and coefficient 8.
$ 5x^3 $ - a term with degree 3 and coefficient 5.

The highest degree term ($ 5x^3 $ in this case) determines the polynomial鈥檚 overall degree, which is 3.

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 each polynomial, determine the mumber of terms and name the coefficients of the terms. $$ x+8 x^{2}+5 x^{3} $$

Short Answer

Step by step solution

Identify the Terms

Count the Terms

Identify Coefficients

Key Concepts

terms in a polynomial

coefficients

polynomial structure

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Probability and Statistics

Statistics

Theoretical and Mathematical Physics

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.