Answer :
The given polynomial function is:
[tex]f(x)=2x^3+3x^2-8x+5[/tex]Since all the co-efficients are integers, we can apply the rational zero theorem.
The trailing co-efficient ( the co-efficient of the constant term) is 5.
Find its factors with the plus and minus sign; thus we have;
[tex]\begin{gathered} \text{Factors}=\pm1,\pm5 \\ \text{These are the possible values for p} \end{gathered}[/tex]The leading co-efficient ( the co-efficient of the term with the highest degree) is 2.
Find its factors with the plus and minus sign; thus we have:
[tex]\begin{gathered} \text{Factors}=\pm1,\pm2 \\ \text{These are the }possible\text{ values for q} \end{gathered}[/tex]Next, is finding all possible values for the rational expression p/q. Thus, we have:
[tex]\begin{gathered} \frac{p}{q}=\pm\frac{1}{1},\pm\frac{1}{2},\pm\frac{5}{1},\pm\frac{5}{2} \\ \frac{p}{q}=\pm1,\pm\frac{1}{2},\pm5,\pm\frac{5}{2} \end{gathered}[/tex]Hence, the possible rational zeros for the polynomial function are:
[tex]\pm1,\pm\frac{1}{2},\pm5,\pm\frac{5}{2}[/tex]