/*! 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 60 For each polynomial function giv... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 60

For each polynomial function given: (a) list each real zero and its multiplicity; (b) determine whether the graph touches or crosses at each \(x\) -intercept; (c) find the \(y\) -intercept and a few points on the graph; (d) determine the end behavior; and (e) sketch the graph. $$f(x)=7 x^{5}-14 x^{4}-21 x^{3}$$

Short Answer

Expert verified
Zeros: 0 (mult. 3), 3, -1; crosses the x-axis. Y-int.: (0,0). End behavior: approaches 鈭� as x鈫掆垶, -鈭� as x鈫�-鈭�.

Step by step solution

01

Factor the polynomial

The given polynomial is \(f(x) = 7x^5 - 14x^4 - 21x^3\). To find the zeros, start by factoring out the greatest common factor (GCF) from each term. The GCF is \(7x^3\), so we have \(f(x) = 7x^3(x^2 - 2x - 3)\). Next, factor \(x^2 - 2x - 3\) as \((x-3)(x+1)\). Thus, the fully factored form is \(f(x) = 7x^3(x-3)(x+1)\).
02

Find the real zeros and their multiplicities

With the factored form \(7x^3(x-3)(x+1)\), identify the zeros by setting each factor to zero:- From \(7x^3 = 0\), the zero is \(x = 0\) with multiplicity 3.- From \(x-3 = 0\), the zero is \(x = 3\) with multiplicity 1.- From \(x+1 = 0\), the zero is \(x = -1\) with multiplicity 1.
03

Determine behavior at each x-intercept

The behavior at each intercept depends on the multiplicity:- At \(x = 0\), the multiplicity is 3 (odd), so the graph crosses the x-axis.- At \(x = 3\), the multiplicity is 1 (odd), so the graph crosses the x-axis.- At \(x = -1\), the multiplicity is 1 (odd), so the graph crosses the x-axis.
04

Find the y-intercept and additional points

To find the y-intercept, set \(x = 0\) in \(f(x)\):\(f(0) = 7(0)^5 - 14(0)^4 - 21(0)^3 = 0\).Therefore, the y-intercept is at \((0, 0)\), which is also an x-intercept.You can find additional points by substituting values for \(x\) to get, for example, \(f(1) = 7(1)^5 - 14(1)^4 - 21(1)^3 = -28\) and \(f(-2) = 7(-2)^5 - 14(-2)^4 - 21(-2)^3 = 84\).
05

Determine the end behavior

The polynomial \(f(x) = 7x^5 - 14x^4 - 21x^3\) is of degree 5 and has a positive leading coefficient. Thus, as \(x\) approaches infinity, \(f(x)\) approaches infinity, and as \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.
06

Sketch the graph

Using the information from previous steps, you can sketch the graph: - Plot the x-intercepts at (0,0), (3,0), and (-1,0), and mark the behavior at these points (crosses the x-axis). - Plot the y-intercept at (0,0). - Use additional points, such as (1,-28) and (-2,84), to help shape the curve. - Indicate end behavior by showing the curve approaching infinity and negative infinity as x moves away from the center of the graph.

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.

Factoring Polynomials
Factoring polynomials is a crucial skill in algebra, allowing us to simplify expressions and solve equations more easily. To factor a polynomial, the first step is to identify the greatest common factor (GCF) among all terms. For the polynomial \(7x^5 - 14x^4 - 21x^3\), the GCF is \(7x^3\). By factoring this out, the polynomial simplifies to \(7x^3(x^2 - 2x - 3)\).
Further factoring of \(x^2 - 2x - 3\) involves recognizing it can be expressed as \((x-3)(x+1)\). The fully factored form is then \(7x^3(x-3)(x+1)\). This representation makes it easier to identify the polynomial's roots and understand its structure.
Real Zeros
Finding the real zeros of a polynomial involves determining the values of \(x\) where the polynomial equals zero. In the factored form \(7x^3(x-3)(x+1)\), each factor of the polynomial can be set to zero:
  • For \(7x^3 = 0\), the zero is \(x = 0\).
  • For \(x-3 = 0\), the zero is \(x = 3\).
  • For \(x+1 = 0\), the zero is \(x = -1\).
These zeros are where the graph intersects or touches the x-axis. Identifying zeros is an essential part of polynomial analysis, providing foundational points for graphing and further analysis.
Multiplicity
The multiplicity of a zero refers to how many times that zero appears as a root of the polynomial. Multiplicity affects the behavior of the graph at the intercepts:
  • For \(x = 0\), the factor \(7x^3\) gives it a multiplicity of 3.
  • For \(x = 3\) and \(x = -1\), they both have a multiplicity of 1.
Generally, if a zero has an odd multiplicity, the graph crosses the x-axis at that intercept. If it's even, the graph merely touches the x-axis.
This specific understanding aids in sketching and predicting the overall shape of the polynomial graph.
End Behavior
End behavior describes how the values of the polynomial function behave as \(x\) approaches positive or negative infinity. For the given polynomial \(f(x) = 7x^5 - 14x^4 - 21x^3\), its leading term is \(7x^5\). Since the highest power of \(x\) is odd, and the leading coefficient is positive, this means:
  • As \(x \to +\infty\), \(f(x) \to +\infty\).
  • As \(x \to -\infty\), \(f(x) \to -\infty\).
Understanding a polynomial's end behavior is crucial for graphing it accurately, as it indicates the direction in which the graph moves.
X-Intercepts
X-intercepts are specific points where the graph of a function crosses the x-axis. For \(f(x) = 7x^3(x-3)(x+1)\), the x-intercepts are at the values of the real zeros:
  • \(x = 0\)
  • \(x = 3\)
  • \(x = -1\)
Each intercept corresponds to a zero of the polynomial. By determining where the graph touches or crosses the x-axis, you can better visualize and also confirm the graph's correctness when plotted.
Y-Intercepts
The y-intercept of a function is found by evaluating the function at \(x = 0\). For our polynomial \(f(x) = 7x^5 - 14x^4 - 21x^3\), setting \(x = 0\) yields:
  • \(f(0) = 0\)
Thus, the y-intercept is at the point \((0,0)\). This point serves as both an x-intercept and a y-intercept in this case, essential for plotting the initial shape of the graph. Identifying the y-intercept helps provide a starting point for understanding the graph's rise or fall as \(x\) changes.

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

For Exercises 77 and 78 , use the following information: One function of particular interest in economics is the profit function. We denote this function by \(P(x)\). It is defined to be the difference between revenue \(R(x)\) and \(\operatorname{cost} C(x)\) so that $$ P(x)=R(x)-C(x) $$ The total revenue received from the sale of \(x\) goods at price \(p\) is given by $$ R(x)=p x $$ The total cost function relates the cost of production to the level of output \(x\). This includes both fixed costs \(C_{\mathrm{f}}\) and variable costs \(C_{\mathrm{r}}\). (costs per unit produced). The total cost in producing \(x\) goods is given by $$ C(x)=C_{\mathrm{f}}+C_{\mathrm{v}} x $$ Thus, the profit function is $$ P(x)=p x-C_{t}-C_{v} x $$ Assume fixed costs are \(\$ 1,000,\) variable costs per unit are \(\$ 20,\) and the demand function is $$ p=100-x $$ Profit. How many units should the company produce to break even?

In Exercises \(55-66\), find the quadratic function that has the given vertex and goes through the given point. vertex: (0,-2)\(\quad\) point: (3,10)

In Exercises \(45-54,\) find the vertex of the parabola associated with each quadratic function. $$f(x)=0.05 x^{2}+2.5 x-1.5$$

Find the asymptotes and intercepts of the rational function \(f(x)=\frac{1}{3 x+1}-\frac{2}{x} .\) (Note: Combine the two expressions into a single rational expression.) Graph this function utilizing a graphing utility. Does the graph confirm what you found?

Suppose the number of individuals infected by a virus can be determined by the formula $$n(t)=\frac{9500 t-2000}{4+t}$$ where \(t>0\) is the time in months. a. Find the number of infected people by the end of the fourth month. b. After how many months are there 5500 infected people? c. What happens with the number of infected people if the trend continues?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure 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.