Final answer:
The student's question involves finding 'coordinates' of a cubic function, which typically means identifying specific points or roots. For a quadratic equation, this can be done using the quadratic formula. However, to find roots of a cubic equation, different methods like factoring, numerical estimation, or the cubic formula are used.
Step-by-step explanation:
The student's question involves solving the polynomial equation f(x) = x^3 + 3x^2 + x + 1 to find its coordinates. However, since this is a cubic equation, it does not have a set of fixed 'coordinates' per se, as it represents a function rather than a point or set of points. To find specific points on the curve of the function, we would need a particular value of x to substitute into the function. If the student was actually asking to find the roots of the polynomial equation (the values of x for which f(x) = 0), this would involve solving the cubic equation, which is a more complex task than solving a quadratic and typically does not have a simple formula like the quadratic formula.
With quadratic equations such as x^2 + 1.2 x 10^-2x - 6.0 × 10^-3 = 0 or x^2 +0.0211x -0.0211 = 0, we can use the quadratic formula to solve for the values of x that make the equation true (the roots). The quadratic formula is derived from the standard form of a quadratic equation, ax^2 + bx + c = 0, and is given by x = (-b ± √(b^2 - 4ac)) / (2a). However, this formula is not applicable directly to cubic equations like the one given the question.
If we misinterpret the initial question as finding the roots rather than the coordinates, we would need to apply different techniques for solving cubic equations, such as factoring (if possible), graphing to estimate the roots, using numerical methods, or even using cubic formulae that are significantly more complicated than the quadratic formula.