- Define the function \(f(x) = -2x^3 + 4x^2 + 6x - 5\).
- Solve the equation \(-2x^3 + 4x^2 + 6x - 5 = 0\) for \(x\).
- Round the resulting roots to the nearest hundredth: \(-1.40, 0.65, 2.76\).
- The zeros are \(\boxed{-1.40, 0.65, 2.76}\).
Explanation
- Analysis of the Function and Goal
We are given the cubic polynomial function \(f(x) = -2x^3 + 4x^2 + 6x - 5\). To find the zeros of the function, we need to solve the equation \(f(x) = 0\), which is:
\[-2x^3 + 4x^2 + 6x - 5 = 0\]
This is a cubic equation, which can have up to three real roots. We will find these roots numerically.
- Calculating the Zeros
By applying numerical methods to solve the cubic equation \(-2x^3 + 4x^2 + 6x - 5 = 0\), we determine the three real roots of the function. The calculated values are approximately:
\[x_1 \approx 2.7589\]
\[x_2 \approx -1.4042\]
\[x_3 \approx 0.6453\]
- Rounding and Final Conclusion
The problem requires us to round these values to the nearest hundredth.
- For \(x_1 \approx 2.7589\), rounding to the hundredths place gives \(2.76\).
- For \(x_2 \approx -1.4042\), rounding to the hundredths place gives \(-1.40\).
- For \(x_3 \approx 0.6453\), rounding to the hundredths place gives \(0.65\).
Thus, the zeros of the polynomial function, rounded to the nearest hundredth, are \(-1.40, 0.65,\) and \(2.76\).
Examples
Finding the roots of a polynomial is essential in many fields like engineering and physics. For example, if you are designing a structural support and the stress distribution follows a cubic curve \(f(x)\), the points where \(f(x) = 0\) tell you where the stress transitions from tension to compression. Identifying these transition points allows engineers to select materials and designs that can safely handle the forces applied to the structure.