Answer
(a) $\pm1,\pm1/3,\pm2,\pm2/3,\pm4,\pm4/3,\pm8,\pm8/3$
(b) 2 or 0 positive, 3 or 1 negative real zeros.
Work Step by Step
(a) $p=\pm1,\pm2,\pm4,\pm8, q=\pm1.\pm3$
$\frac{p}{q}=\pm1,\pm1/3,\pm2,\pm2/3,\pm4,\pm4/3,\pm8,\pm8/3$
(b) $P(x)$ has 2 sign changes, meaning possible 2 or 0 positive real zeros.
$P(-x)=-3^7+x^5+5x^4-x^3+8$ which has 3 sign changes, meaning possible
3 or 1 negative real zeros.
