Final answer:
To factor and solve the polynomial equation f(x) = x^4 - x^3 - 52x^2 - 62x + 240, one typically needs to use numerical methods or graphing to find approximations of the roots, as it doesn't factor readily. For quadratic equations, like a simplified example x² + 0.0211x - 0.0211 = 0, the quadratic formula would be used to find the solutions.
Step-by-step explanation:
To factor and solve the equation f(x) = x^4 – x^3 – 52x^2 – 62x + 240, we need to find values of x that make the equation equal to zero. This process usually involves factoring the polynomial to find the roots, which can sometimes be done by grouping terms or using synthetic division. However, for higher-degree polynomials like this one, factoring by inspection can be quite difficult, and we might need to use numerical methods or graphing techniques to approximate the solutions.
For quartic equations that don't factor easily, we might use techniques such as Descartes' Rule of Signs to determine the number of positive and negative real roots or graph the function to find where it intersects the x-axis. Unfortunately, without a straightforward method to factor this quartic directly in the question provided, we're unable to solve it completely.
If the polynomial were quadratic instead, such as x² + 0.0211x - 0.0211 = 0, we would solve it using the quadratic formula as follows: x = (-b ± √(b² - 4ac)) / (2a), where a is the coefficient before x², b is the coefficient before x, and c is the constant term.