- Identify the quartic polynomial \(f(x) = -4x^4 - 7x^3 + 9x^2 + 11x - 3\).
- Find the real roots by solving \(f(x) = 0\), resulting in approximately \(-2.1209\), \(-1.1574\), \(0.2365\), and \(1.2918\).
- Round each root to the nearest hundredth.
- The final roots are \(\boxed{-2.12, -1.16, 0.24, 1.29}\).
Explanation
- Problem Analysis
The problem asks us to find all the real roots of the quartic polynomial function given by \(f(x) = -4x^4 - 7x^3 + 9x^2 + 11x - 3\). A root of the function is any real value \(x\) that satisfies the equation \(f(x) = 0\). As this is a polynomial of degree 4, we expect at most 4 real roots. To find these values, we solve the equation \(-4x^4 - 7x^3 + 9x^2 + 11x - 3 = 0\).
- Calculating the Roots
We can determine the numerical values of the roots by applying numerical methods suitable for finding the zeros of a polynomial. Upon solving the equation \(-4x^4 - 7x^3 + 9x^2 + 11x - 3 = 0\), we identify the four roots as approximately \(1.29179552\), \(-2.12090354\), \(-1.15740779\), and \(0.23651581\).
- Rounding and Finalizing the Roots
The problem requires us to round each of the identified roots to the nearest hundredth.
- For \(x_1 \approx 1.29179552\), the nearest hundredth is \(1.29\).
- For \(x_2 \approx -2.12090354\), the nearest hundredth is \(-2.12\).
- For \(x_3 \approx -1.15740779\), the nearest hundredth is \(-1.16\).
- For \(x_4 \approx 0.23651581\), the nearest hundredth is \(0.24\).
Sorting these values from smallest to largest, we obtain \(-2.12, -1.16, 0.24, 1.29\). Thus, the roots of the function are \(-2.12, -1.16, 0.24, 1.29\).
Examples
Polynomial functions are fundamental in modeling real-world phenomena. For example, if a company models its profit based on the quantity of items produced, the roots of the resulting profit function represent the 'break-even' points—the exact production levels where the company makes zero profit, transitioning from a loss to a gain or vice versa. Finding these roots helps managers understand the range of production that leads to profitability.