- Define the polynomial function \(f(x) = x^3 - 5x^2 + 2x + 6\).
- Evaluate the function for each candidate value: \(f(-0.9) \approx -0.58\), \(f(0.8) \approx 4.91\), \(f(4.2) \approx 0.29\), \(f(1.7) \approx -0.14\), and \(f(6.0) = 54\).
- Select the three values for which \(f(x)\) is closest to 0.
- The correct solutions are \(\boxed{-0.9, 4.2, 1.7}\).
Explanation
- Problem Analysis and Objective
We are tasked with finding the three roots of the cubic equation \(f(x) = x^3 - 5x^2 + 2x + 6 = 0\) from a given list of five candidates: \(-0.9\), \(0.8\), \(4.2\), \(1.7\), and \(6.0\). A value is considered a root if it satisfies the equation, meaning \(f(x) \approx 0\). We will evaluate the polynomial \(f(x)\) for each candidate to determine which ones result in a value closest to zero.
- Evaluating the Polynomial for Given Candidates
We calculate the value of the function \(f(x) = x^3 - 5x^2 + 2x + 6\) for each given value:
- For \(x = -0.9\): \(f(-0.9) = (-0.9)^3 - 5(-0.9)^2 + 2(-0.9) + 6 = -0.729 - 4.05 - 1.8 + 6 = -0.579\).
- For \(x = 0.8\): \(f(0.8) = (0.8)^3 - 5(0.8)^2 + 2(0.8) + 6 = 0.512 - 3.2 + 1.6 + 6 = 4.912\).
- For \(x = 4.2\): \(f(4.2) = (4.2)^3 - 5(4.2)^2 + 2(4.2) + 6 = 74.088 - 88.2 + 8.4 + 6 = 0.288\).
- For \(x = 1.7\): \(f(1.7) = (1.7)^3 - 5(1.7)^2 + 2(1.7) + 6 = 4.913 - 14.45 + 3.4 + 6 = -0.137\).
- For \(x = 6.0\): \(f(6.0) = (6.0)^3 - 5(6.0)^2 + 2(6.0) + 6 = 216 - 180 + 12 + 6 = 54\).
- Identifying the Solutions and Conclusion
By comparing the results of \(f(x)\), we look for the values that are closest to zero. The function values are:
- \(f(-0.9) \approx -0.58\)
- \(f(0.8) \approx 4.91\)
- \(f(4.2) \approx 0.29\)
- \(f(1.7) \approx -0.14\)
- \(f(6.0) = 54\)
The values of \(f(x)\) closest to zero are at \(x = -0.9\), \(x = 4.2\), and \(x = 1.7\). These three candidates represent the roots of the cubic equation to the nearest tenth. Therefore, the solutions from the provided list are \(x = -0.9\), \(x = 4.2\), and \(x = 1.7\).
Examples
In engineering, finding the roots of cubic equations is a common task when determining the stability or intersection points of non-linear systems. For example, when analyzing the trajectory of a projectile subject to drag, the distance or time it takes to reach a specific altitude often involves solving cubic polynomials. Identifying these roots allows engineers to predict when an object will reach a target height or hit the ground.