Final answer:
To approximate the real zeros of the function f(x), we can use graphing calculators or software that find where the function crosses the x-axis, or apply numerical root-finding methods. Some of these roots might be complex.
Step-by-step explanation:
To approximate the real zeros of the polynomial function f(x) = 6x⁴ - 5x³ + 2x² - 5x - 4, we use various methods. The most common are the Rational Roots Theorem, synthetic division, and graphing calculators or computer algebra systems. Since the question asks for an approximation to the nearest tenth, using a graphing method can be a good option.
For example, we can graph the function using a calculator and look for where the graph crosses the x-axis. These points are our real zeros. If the graph is not available or if we prefer an analytical approximation, we can use numerical methods such as the Newton-Raphson method or software tools that can apply sophisticated root-finding algorithms.
It's important to note that not all roots of a quartic function may be real; some may be complex. If our function has real rational roots, they will be of the form p/q, where p is a factor of the constant term (-4) and q is a factor of the leading coefficient (6).