Find all the rational zeros. Write the answer in exact form. \[ q(x)=3 x^{4}-2 x^{3}-7 x^{2}+4 x+2 \] If there is more than one answer, separate them with commas. Select "None" if applicable. The rational zeros of \( q(x): \square \).

To find the rational zeros of the polynomial \( q(x) = 3x^4 - 2x^3 - 7x^2 + 4x + 2 \), we can apply the Rational Root Theorem. This theorem suggests that potential rational roots are of the form ± factor of the constant term divided by ± factor of the leading coefficient. The constant term is 2 (factors: ±1, ±2) and the leading coefficient is 3 (factors: ±1, ±3). Thus, the possible rational roots are ±1, ±2, ±1/3, ±2/3. Testing these values, we find: - \( q(1) = 3(1)^4 - 2(1)^3 - 7(1)^2 + 4(1) + 2 = 0 \) This means \( x = 1 \) is a rational zero. Next, we can use synthetic division to factor \( q(x) \) and simplify our search for other potential zeros. We find that dividing \( q(x) \) by \( x - 1 \) gives us a cubic polynomial. By further checking possible rational roots of this cubic, we also find \( x = -\frac{2}{3} \) and \( x = 1 \) again until we factor completely. The rational zeros of \( q(x) \) are \( 1, -\frac{2}{3}, \) and \( \frac{1}{3} \), which gives us the final results. So, the rational zeros of \( q(x): 1, -\frac{2}{3}, \frac{1}{3}. \)