/*! 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 4 In Exercises \(1-10\), determine... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 4

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x$$

Short Answer

Expert verified
The function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\) is a polynomial function and its degree is 7.

Step by step solution

01

Determine if the function is a polynomial

From the definition of a polynomial, we see that our function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\) involves only addition, multiplication and non-negative integer exponents of x, which are 7, 5, and 1. There aren't any variables under a square root, in the denominator, or inside a trigonometric or logarithmic function. We don't see that any coefficients are complex numbers, all of them are real numbers in this case. Thus, we can confirm the function g(x) is a polynomial function.
02

Identify the degree of the polynomial

The degree of a polynomial function is the highest power of the variable (x) in the polynomial. Looking at the polynomial function \(g(x)=6 x^{7}+\pi x^{5}+\frac{2}{3} x\), the highest power of x is 7. Thus, the degree of the polynomial is 7.

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.

degree of a polynomial
The degree of a polynomial is a critical aspect of understanding polynomial functions. It can give us insight into the function's behavior and characteristics as graphing and solving it becomes involved. The degree of a polynomial is defined as the highest power of the variable present when the polynomial is expressed in its standard form.
\[p(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0\]
In this equation, the degree of the polynomial is \(n\) because \(n\) is the highest exponent on the variable \(x\).

  • For instance, in the polynomial, \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), the highest power of \(x\) is 7, making the degree of this polynomial 7.
  • The degree tells us the number of roots or solutions the polynomial can possibly have.
  • The degree influences the end behavior of the polynomial function graph.
Being able to identify the degree of the polynomial is foundational to analyzing polynomials in mathematics.
real coefficients
Coefficients are the numbers in front of the variables in a polynomial, and when we discuss real coefficients, we mean these numbers belong to the set of real numbers. Real numbers include all rational and irrational numbers, encompassing integers, fractions, and non-fractional numbers with decimal expansions that neither repeat nor terminate, like \(\pi\) or square roots of non-perfect squares.
When a polynomial has real coefficients, like \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), we ensure that each coefficient, such as 6, \(\pi\), and \(\frac{2}{3}\), is a real number.

  • Real coefficients guarantee that the polynomial expression is defined for all real numbers \(x\).
  • They allow the polynomial to be graphed on the Cartesian plane smoothly.
  • Complex coefficients, involving \(i\), are not present here, simplifying analysis and calculations.
Real coefficients keep the polynomial firmly within the realm of real and practical applications.
non-negative integer exponents
Within the definition of a polynomial, using non-negative integer exponents is essential. This means that the exponents on the variable \(x\) must be whole numbers starting from zero upwards: 0, 1, 2, and so on. Using exclusively non-negative integer exponents avoids the complications that come with fractions or negative numbers, which can introduce undefined or complex scenarios.
In the polynomial function \(g(x) = 6x^7 + \pi x^5 + \frac{2}{3}x\), the exponents are all non-negative integers: 7, 5, and 1.

  • These exponents ensure the polynomial is continuous and smooth over all real numbers.
  • They prevent the introduction of mathematical operations such as division by zero or taking roots of negative numbers.
  • Polynomials with non-negative integer exponents are easily differentiated and integrated, which is useful in calculus.
Non-negative integer exponents form the foundation of polynomials, ensuring they are straightforward and tractable.

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

Give an example of a function that is not subject to the Intermediate Value Theorem.

Which one of the following is true? a. The function \(f(x)=\frac{1}{\sqrt{x-3}}\) is a rational function. b. The \(x\) -axis is a horizontal asymptote for the graph of $$f(x)=\frac{4 x-1}{x+3}.$$ c. The number of televisions that a company can produce per week after \(t\) weeks of production is given by $$N(t)=\frac{3000 t^{2}+30,000 t}{t^{2}+10 t+25}.$$ Using this model, the company will eventually be able to produce \(30,000\) televisions in a single week. d. None of the given statements is true.

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-x^{2}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. The volume of a gas varies directly as its temperature and inversely as its pressure. At a temperature of 100 Kelvin and a pressure of 15 kilograms per square meter, the gas occupies a volume of 20 cubic meters. Find the volume at a temperature of 150 Kelvin and a pressure of 30 kilograms per square meter.

Use the four-step procedure for solving variation problems given on page 356 to solve. The intensity of illumination on a surface varies inversely as the square of the distance of the light source from the surface. The illumination from a source is 25 foot-candles at a distance of 4 feet. What is the illumination when the distance is 6 feet?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

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.