- The function \(f(x) = x^4 - 2x^3 - 61x^2 + 62x + 840\) is a fourth-degree polynomial with four roots.
- We test potential rational roots using the factors of 840 and verify that \(-6, -4, 5,\) and \(7\) satisfy the equation \(f(x) = 0\).
- Arranging the roots in ascending order gives the final result.
- The zeros of the function are \(\boxed{-6,-4,5,7}\).
Explanation
- Analyzing the Polynomial
To find the zeros of the polynomial \(f(x) = x^4 - 2x^3 - 61x^2 + 62x + 840\), we look for values of \(x\) such that \(f(x) = 0\). Since this is a fourth-degree polynomial, it can have up to four real roots. According to the Rational Root Theorem, any potential rational root must be a factor of the constant term, 840. Some factors of 840 include \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 7, \pm 8, \dots\). We can test these values to identify the zeros.
- Identifying the Roots
We evaluate the polynomial at various integer factors to find the roots.
Testing \(x = -6\):
\[f(-6) = (-6)^4 - 2(-6)^3 - 61(-6)^2 + 62(-6) + 840 = 1296 + 432 - 2196 - 372 + 840 = 0\]
Testing \(x = -4\):
\[f(-4) = (-4)^4 - 2(-4)^3 - 61(-4)^2 + 62(-4) + 840 = 256 + 128 - 976 - 248 + 840 = 0\]
Testing \(x = 5\):
\[f(5) = (5)^4 - 2(5)^3 - 61(5)^2 + 62(5) + 840 = 625 - 250 - 1525 + 310 + 840 = 0\]
Testing \(x = 7\):
\[f(7) = (7)^4 - 2(7)^3 - 61(7)^2 + 62(7) + 840 = 2401 - 686 - 2989 + 434 + 840 = 0\]
Since we have found four distinct roots for a fourth-degree polynomial, we have identified all the zeros of the function.
- Final Result
The zeros of the function \(f(x) = x^4 - 2x^3 - 61x^2 + 62x + 840\) are \(-6, -4, 5,\) and \(7\). Arranging them from smallest to largest, we obtain \(-6, -4, 5, 7\). The final answer is \(\boxed{-6,-4,5,7}\).
Examples
Finding the roots of a polynomial is essential in many engineering and scientific fields. For example, in control systems engineering, the roots of a characteristic equation (the eigenvalues) determine the stability of a system. By finding the zeros, an engineer can predict how a system will behave over time, such as whether it will oscillate or settle into a steady state. Similarly, in economics, finding the roots of a supply-demand function can help identify equilibrium points where production perfectly matches consumer desire.