Since the polynomial g(x) is a polynomial with integer coefficients and if p/q is a zero of:
[tex]\begin{gathered} G(x)(G(\frac{p}{q})=0),\text{ then p is a factor of the constant term of }G(x)\text{ and q} \\ is\text{ a factor of the leading coefficient} \end{gathered}[/tex]So, we have to find all integer factor of the constant term, in this case: -4 and the leading coefficien, in this case: 5, and find all possible quotients p/q where p is a factor of the constant and q is the factor of the leading coefficient:
[tex]\begin{gathered} \text{Integers factor of -4: }\pm1,\text{ }\pm2,\text{ }\pm4 \\ \text{Integers factor of 5: }\pm1,\text{ }\pm5 \\ \text{Then, the possible rational roots are:} \\ \pm1,\pm2,\pm4,\pm5,\pm\frac{4}{5},\text{ }\pm\frac{1}{5},\text{ }\pm\frac{2}{5} \end{gathered}[/tex]To find which of these are roots, simply substituing on the original equation:
By substitung all the values of possible roots, we can state that this polynomial doesnt' have rational zeros.
