- Analyze the polynomial \(f(x) = -4x^4 - 7x^3 + 9x^2 + 11x - 3\).
- Calculate the numerical roots of the function.
- Round each root to the nearest hundredth.
- The real zeros are \(\boxed{-2.12, -1.16, 0.24, 1.29}\).
Explanation
- Analysis of the polynomial function
We analyze the polynomial function \(f(x) = -4x^4 - 7x^3 + 9x^2 + 11x - 3\). This is a fourth-degree polynomial, which means it can have at most 4 real roots. To find the zeros, we need to determine the values of \(x\) for which \(f(x) = 0\). We can numerically find these roots by solving the equation \(-4x^4 - 7x^3 + 9x^2 + 11x - 3 = 0\).
- Finding the zeros of the function
By performing numerical calculations on the given polynomial, we identify the four real roots of the function. The values obtained are approximately \(x \approx 1.2918\), \(x \approx -2.1209\), \(x \approx -1.1574\), and \(x \approx 0.2365\). These represent the points where the graph of the function crosses or touches the x-axis.
- Rounding and final results
Rounding each of the identified zeros to the nearest hundredth, we get:
- \(1.2918 \approx 1.29\)
- \(-2.1209 \approx -2.12\)
- \(-1.1574 \approx -1.16\)
- \(0.2365 \approx 0.24\)
The zeros of the polynomial are \(-2.12, -1.16, 0.24, 1.29\).
Examples
Polynomial functions are widely used in engineering and physics to model paths or changes. For example, finding the zeros of a polynomial can help determine when a projectile launched into the air will return to the ground (at height \(h=0\)). By modeling the height of the projectile as a function of time \(t\), solving for the zeros of the function allows us to find the exact moments when the object hits the ground, providing crucial information for safety and flight planning.