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Chapter 3: Problem 49

Use a graphical method to find all real solutions of each equation. Express solutions to the nearest hundredth. $$0.86 x^{3}-5.24 x^{2}+3.55 x+7.84=0$$

Short Answer

Expert verified
Graph the equation and identify the x-intercepts to find the solutions to the nearest hundredth.

Step by step solution

01

Graph the Equation

First, use a graphing calculator or software to plot the graph of the function \( f(x) = 0.86x^3 - 5.24x^2 + 3.55x + 7.84 \). This visual representation will allow you to identify the x-values where the function crosses the x-axis.
02

Locate the x-intercepts

Examine the graph to find the points where the curve intersects the x-axis. These points are your potential real solutions to the equation \( f(x) = 0 \).
03

Determine Accuracy of Solutions

Zoom in on each intersection point on the graph to refine the x-value estimates. Adjust the window or use tracing or calculating tools from the software to pinpoint x-intercepts with greater precision.
04

Record the Solutions

Record the x-values at the intersections of the graph with the x-axis, ensuring they are expressed to the nearest hundredth. These values are your real solutions to the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding X-Intercepts
When working with polynomial equations, one of the key steps in identifying real solutions is locating the x-intercepts of the graph. But what exactly are x-intercepts? Simply put, an x-intercept is a point where the graph of a function crosses the x-axis. This means at this point, the value of the function, denoted as \( f(x) \), is zero.
  • This intersection happens because the polynomial equation equals zero at the x-intercept.
  • The x-coordinates of these points are your potential real solutions.
  • Finding x-intercepts directly correlates with solving the equation \( f(x) = 0 \).
Keep in mind, a graph can have multiple x-intercepts. For polynomial functions, the number of real x-intercepts is determined by the degree of the equation, meaning that a cubic equation like ours can have up to three real solutions.
Using a Graphing Calculator
Graphing calculators are powerful tools for visualizing functions and finding solutions. To identify the x-intercepts of a polynomial equation using a graphing calculator, follow these steps:
  • Input the polynomial equation into the calculator, ensuring all coefficients and powers are correctly entered.
  • Use the graphing feature to plot the function, which will give you a visual representation of the mathematical curve.
  • Look for where this curve crosses the x-axis, as these points represent your x-intercepts.
Most calculators allow you to adjust the viewing window and zoom in, helping to estimate the x-intercepts accurately. Some models even have a 'trace' feature to better locate these points and refine your solution further to the nearest hundredth as required.
Exploring Polynomial Equations
Polynomial equations are expressions involving a sum of powers in one or more variables multiplied by coefficients. The equation from the exercise, \(0.86x^3 - 5.24x^2 + 3.55x + 7.84 = 0\) is a cubic polynomial because the highest power of \(x\) is 3.
  • A cubic equation can have multiple roots, ranging from one to three real roots.
  • The solutions, or roots, of the equation are values of \(x\) that make the equation true, meaning they satisfy \( f(x) = 0 \).
  • In graphical terms, these are the x-intercepts of the function.
Understanding the general behavior and characteristics of polynomial functions helps predict and interpret the number and type of solutions without needing to perform extensive calculations.
Identifying Real Solutions
Real solutions to a polynomial equation are values of \(x\) for which the equation \( f(x) = 0 \). Using the graphical approach to find these solutions involves observing where the polynomial's graph intersects the x-axis.
  • These points on the graph represent real number solutions because they are x-values where the function is equal to zero.
  • Zooming in on the graph allows one to identify these solutions accurately to the nearest hundredth, as specified in many exercises.
  • A graph may suggest a number of real solutions based on its intersections; these need to be verified for precision.
By understanding graphical methods and their role in identifying real solutions, students can grasp polynomial behaviors more comprehensively.

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Most popular questions from this chapter

For each polynomial function, do the following in order. (a) Use Descartes' rule of signs to find the possible number of positive and negative real zeros. (b) Use the rational zeros theorem to determine the possible rational zeros of the function. (c) Find the rational zeros, if any. (d) Find all other real zeros, if any. (e) Find any other nonreal complex zeros, if any. (f) Find the \(x\) -intercepts of the graph, if any. (g) Find the \(y\) -intercept of the graph. (h) Use synthetic division to find \(P(4),\) and give the coordinates of the corresponding point on the graph. (i) Determine the end behavior of the graph. (i) Sketch the graph. (You may wish to support your answer with a calculator graph.) $$P(x)=3 x^{4}-14 x^{2}-5$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}-8 x^{2}+16 \\\&=(x+2)^{2}(x-2)^{2}\end{aligned}$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=24 x^{3}+40 x^{2}-2 x-12$$

Factor \(P(x)\) into linear factors given that \(k\) is a zero of \(P\). $$P(x)=-6 x^{3}-17 x^{2}+63 x-10 ; \quad k=-5$$

Use the rational zeros theorem to factor \(P(x)\). $$P(x)=12 x^{3}+40 x^{2}+41 x+12$$
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