A polynomial with integer coefficients is of the form
\[21x^4 + a_3 x^3 + a_2 x^2 + a_1 x - 28 = 0.\]
If $r$ is a rational root of this polynomial, then find the number of different possible values of $r.$
If $r$ is a rational root of the given polynomial, then by the Rational Root Theorem, $r$ must be a divisor of 28. Therefore, the possible values of $r$ are $\pm 1,$ $\pm 2,$ $\pm 4,$ $\pm 7,$ $\pm 14,$ and $\pm 28,$ for a total of $2 \cdot 6 = \boxed{12}.$
To find the possible rational roots of the polynomial, we can use the Rational Root Theorem. According to the theorem, if $r$ is a rational root of a polynomial with integer coefficients, then $r$ must be of the form $\frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.
In this case, the constant term is $-28$ and the leading coefficient is $21$. So, the possible rational roots are of the form $\frac{p}{q}$, where $p$ is a factor of $-28$ and $q$ is a factor of $21$.
The factors of $-28$ are: $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$.
The factors of $21$ are: $\pm 1, \pm 3, \pm 7, \pm 21$.
Combining these factors, we get the possible rational roots as: $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{7}{3}, \pm \frac{14}{3}, \pm \frac{28}{3}$.
Therefore, there are a total of $\boxed{24}$ different possible values of $r$.
To find the number of different possible values of $r,$ we can use the Rational Root Theorem.
The Rational Root Theorem states that if a polynomial with integer coefficients has a rational root $\frac{p}{q}$ (where $p$ and $q$ are integers with no common factors other than 1) and $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient, then $\frac{p}{q}$ is a possible root of the polynomial.
In this case, the polynomial has a constant term of $-28$ and a leading coefficient of $21.$ Therefore, the possible rational roots can be determined by listing all the factors of $-28$ (the constant term) divided by all the factors of $21$ (the leading coefficient).
The factors of $-28$ are $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28.$
The factors of $21$ are $\pm 1, \pm 3, \pm 7, \pm 21.$
Combining these factors, we get the following possible rational roots:
$\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28,$ and $\frac{\pm 1}{\pm 1}, \frac{\pm 2}{\pm 1}, \frac{\pm 4}{\pm 1}, \frac{\pm 7}{\pm 1}, \frac{\pm 14}{\pm 1}, \frac{\pm 28}{\pm 1}.$
Simplifying these fractions, we have $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{4}{1}, \pm \frac{7}{1}, \pm \frac{14}{1}, \pm \frac{28}{1}.$
Therefore, there are a total of $24$ different possible rational roots for this polynomial.