To find the $x$-intercepts of the polynomial function $f(x) = x^3 + 0.1x^2 - 1.6279x - 0.31222$, we need to solve for $x$ when $f(x) = 0$. This involves finding the roots of the cubic equation:

$x^3 + 0.1x^2 - 1.6279x - 0.31222 = 0$

Since this is a cubic equation, we can either attempt to factor it (if possible) or use numerical methods, such as the Rational Root Theorem, graphing, or a root-finding algorithm (like Newton's method), to approximate the roots.

Let's find the approximate roots of the equation using numerical techniques.

Solving this equation numerically:

The $x$-intercepts are approximately:

$x \approx -1.55, 0.20, 1.76$

These are the points where the function crosses the $x$-axis. Therefore, the $x$-intercepts are:

$\boxed{-1.55, 0.20, 1.76}$

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Would you like more details on how these were calculated, or any other part of this problem?

Here are 5 related questions to expand on this:

1. How would you find the y-intercept of a cubic function?
2. Can a cubic function have more than three real roots?
3. What is the significance of the discriminant in determining the nature of cubic roots?
4. How does the Rational Root Theorem help in finding roots of polynomials?
5. Can you explain the graphical behavior of a cubic function near its intercepts?

Tip: When solving cubic equations, graphing can provide insights into the approximate locations of the roots before applying numerical methods!