/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 53 OPEN ENDED. Write a polynomial o... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 53

OPEN ENDED. Write a polynomial of degree 5 that has three terms.

Short Answer

Expert verified
A polynomial could be \(2x^5 - 3x^3 + 4x^2\).

Step by step solution

01

Understand the Requirements

A polynomial of degree 5 should have its highest exponent as 5. It needs to contain three distinct terms.
02

Choose a Leading Term

Select a term with the variable raised to the fifth power, such as \(2x^5\). This is the leading term of our polynomial.
03

Add Additional Terms

Choose two more terms with different exponents for variety, like \(-3x^3\) and \(+4x^2\). These terms ensure the polynomial has three terms.
04

Combine the Terms

Combine all the terms to form a single polynomial: \(2x^5 - 3x^3 + 4x^2\).

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.

Leading Term
The leading term in a polynomial is the term with the highest degree, meaning it has the largest exponent of the variable. In a polynomial of degree 5, the leading term is crucial as it determines the degree of the entire expression. For example, if we have the polynomial \(2x^5 - 3x^3 + 4x^2\), the leading term is \(2x^5\). This term is important because:
  • It influences the end behavior of the polynomial graph.
  • The coefficient (here, 2) affects how steep or flat the graph appears.
The degree of the leading term being 5 indicates that the polynomial itself is of degree 5 because it holds the highest power of \(x\). Understanding the role of the leading term helps in grasping both the behavior and classification of polynomials.
Polynomial Expression
A polynomial expression is a mathematical expression consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are like building blocks in algebra and come in various forms, like monomials, binomials, trinomials, and so forth.
  • A polynomial of degree 5 includes terms where the highest exponent is 5.
  • In the expression \(2x^5 - 3x^3 + 4x^2\), each term is part of the polynomial structure.
  • Such an expression typically consists of several terms, each with its coefficient and exponent.
When writing a polynomial of degree 5, one should ensure that the highest power of the variable is 5, while other terms may vary in their powers and coefficients according to the problem's requirements.
Exponents
Exponents are a shorthand way to express repeated multiplication of a number by itself. In polynomials, they reveal how many times a base number is used as a factor. For instance, \(x^5\) means \(x\) is multiplied by itself five times. Exponents play a vital role in:
  • Determining the polynomial's degree: the highest exponent indicates the maximum possible degree.
  • Influencing the shape and complexity of the polynomial function's graph.
For our example polynomial, \(2x^5 - 3x^3 + 4x^2\), the exponents are 5, 3, and 2. This defines how each term is structured. By understanding exponents, we can grasp how different terms within a polynomial interact to shape the overall expression. Careful use of exponents allows for diverse polynomial functions.

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{5 y+2 z=11} \\ {10 y-4 z=-2}\end{array} $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}+k x-17\right) \div(x-2) $$

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ p\left(x^{2}+4\right) $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \left[\begin{array}{rr}{3} & {6} \\ {2} & {-1}\end{array}\right] \cdot\left[\begin{array}{l}{a} \\\ {b}\end{array}\right]=\left[\begin{array}{c}{-3} \\ {18}\end{array}\right] $$

Solve each matrix equation or system of equations by using inverse matrices. $$ \begin{array}{l}{3 j+2 k=8} \\ {j-7 k=18}\end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.