DEFINITIONS
The Rational Roots Test (also known as Rational Zeros Theorem) allows us to find all possible rational roots of a polynomial. Suppose we have some polynomial P(x) with integer coefficients and a nonzero constant term, the possible rational roots are given in the form:
[tex]\pm\frac{p}{q}[/tex]where p represents the factors of the constant term of the polynomial and q represents the factors of the leading coefficient.
SOLUTION
The polynomial is given to be:
[tex]3x^2+2x+2[/tex]The leading coefficient is 3 and the constant term is 2.
Since all coefficients are integers, we can apply the rational zeros theorem.
The factors of the leading coefficient are 1 and 3, while the factors of the constant term are 1 and 2. Therefore, we have that:
[tex]\begin{gathered} p=\pm1,\pm2 \\ q=\pm1,\pm3_{} \end{gathered}[/tex]Hence, the possible roots are:
[tex]\begin{gathered} \frac{p}{q}=\pm\frac{1}{1},\pm\frac{1}{3},\pm\frac{2}{1},\pm\frac{2}{3}_{} \\ \therefore \\ \frac{p}{q}=\pm1,\pm\frac{1}{3},\pm2,\pm\frac{2}{3} \end{gathered}[/tex]