A polynomial with integer coefficients is of the form

Question

\[21x^4 + a_3 x^3 + a_2 x^2 + a_1 x - 28 = 0.\]
If $r$ is a rational root of this polynomial, then find the number of different possible values of $r.$

Answer 1

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 $2 \cdot 6 = \boxed{12}.$