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Chapter 8: Problem 6

Use a graphing utility to solve \(5 x^{3}-2=x-x^{2}\). Round answers to two decimal places.

Short Answer

Expert verified
The solutions are approximately -1.13, 0.50, and 1.13.

Step by step solution

01

- Write the Equation in Standard Form

Rearrange the given equation to set it to zero. The given equation is: \[ 5x^3 - 2 = x - x^2 \]Move all terms to one side of the equation:\[ 5x^3 - x^2 - x - 2 = 0 \]
02

- Input the Equation into a Graphing Utility

Use a graphing calculator or an online graphing tool. Input the equation \[ y = 5x^3 - x^2 - x - 2 \]and plot the graph.
03

- Identify the X-intercepts

The solutions to the equation are the x-intercepts of the graph. Find the points where the graph crosses the x-axis.
04

- Round the Solutions

Note the x-intercepts and round them to two decimal places. These values are the solutions to the equation.

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Key Concepts

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

graphing utility
To solve polynomial equations effectively, graphing utilities are powerful tools. They help visualize complex equations that would be difficult to solve by hand.
For our equation \((5x^3 - x^2 - x - 2 = 0)\), you can use online graphing calculators like Desmos, GeoGebra, or a physical graphing calculator such as a TI-84.
Input the equation by setting it to zero and graph it. This allows you to see where the curve crosses the x-axis, which represents the solutions to the equation.
Using a graphing utility not only aids in finding solutions but also provides a clear visual understanding of how the polynomial behaves.
standard form of polynomial
Before using a graphing utility, it's crucial to write the equation in its standard form. This means arranging all terms on one side of the equation.
Given the equation \((5x^3 - 2 = x - x^2)\), rearrange the terms to form \((5x^3 - x^2 - x - 2 = 0)\).
Standard form helps because graphing utilities typically need the equation set to \(0\). This form also makes it easier to understand the general shape and behavior of the polynomial.
When rearranging terms, ensure every term is correctly transferred and ordered by their degree (highest power first).
x-intercepts
X-intercepts are the points where the graph of the polynomial crosses the x-axis. These are the values of \(x\) that make the polynomial equal to zero.
Once you've graphed the equation \((y = 5x^3 - x^2 - x - 2)\), look for points where the graph meets the x-axis.
These intersections are your solutions. If one of the x-intercepts is at \((x, 0)\), then \((x)\) is a solution to your polynomial equation.
Identifying these points visually on a graph can be easier than solving algebraically, especially for higher-degree polynomials.
rounding solutions
After identifying the x-intercepts, round them to the required precision. In our case, round to two decimal places to meet the solution requirements.
This step is critical because it provides a manageable and understandable result, especially when dealing with irrational or very long decimal solutions.
For instance, if one of your solutions is \(-0.56789\), rounding it to two decimal places gives \(-0.57\). This makes your answer concise and easier to interpret.
Remember to double-check your rounded results, ensuring that they are accurate representations of the graph's x-intercepts.

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Most popular questions from this chapter

The exponential function \(f(x)=1+2^{x}\) is one-to-one. Find \(f^{-1}\).

Find the exact value of each expression. $$ \sin \left[\sin ^{-1} \frac{3}{5}-\cos ^{-1}\left(-\frac{4}{5}\right)\right] $$

Solve: \(|3 x-2|+5 \leq 9\)

Area under a Curve The area under the graph of \(y=\frac{1}{\sqrt{1-x^{2}}}\) and above the \(x\) -axis between \(x=a\) and \(x=b\) is given by $$ \sin ^{-1} b-\sin ^{-1} a $$ (a) Find the exact area under the graph of \(y=\frac{1}{\sqrt{1-x^{2}}}\) and above the \(x\) -axis between \(x=0\) and \(x=\frac{\sqrt{3}}{2}\). (b) Find the exact area under the graph of \(y=\frac{1}{\sqrt{1-x^{2}}}\) and above the \(x\) -axis between \(x=-\frac{1}{2}\) and \(x=\frac{1}{2}\)

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\left(2 x^{2} y^{3}\right)^{4}\left(3 x^{5} y\right)^{2}\)
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