Answer
please see "step by step"
Work Step by Step
1. Start by listing the candidates for rational zeros (rational numbers $\displaystyle \frac{p}{q}$ ) by determining possible
1.1. p's: list all ($\pm$)factors of the constant term
1.2. q's: list all ($\pm$)factors of the coefficient of the leading term,
1.3. $\displaystyle \frac{p}{q}:$ list all possible different fractions (rational numbers) using the p's and q's.
2. Test (one by one) rational numbers $\displaystyle \frac{p}{q}$ from 1.3 by dividing the polynomial by ($x-\displaystyle \frac{p}{q}).$
(usually by synthetic division).\ "Test" means "check if the remainder is 0"
3. For candidates $\displaystyle \frac{p}{q}$ that pass the test in 2, see if the quotient can be factored.
If it can, fully factor the polynomial and use the zero product principle to determine the zeros.
If you can not factor the quotient, repeat steps 2 and 3, but instead of the initial polynomial, divide the last quotient with $(x-\displaystyle \frac{p}{q}).$
4. Once all the $\displaystyle \frac{p}{q}$ from 1.3 are exhausted (note: some may be repeated zeros), you will have completed the task.
You may have a left-over quotient that needs to be factored, but its factoring will not yield any further RATIONAL zeros (since all possible candidates have been tested).
