Final answer:
To find rational zeros of a polynomial function, utilize the Rational Root Theorem to generate a list of potential zeros, test them through substitution or synthetic division, and iteratively factor the polynomial.
Step-by-step explanation:
To find all the rational zeros of a polynomial function, one can follow a process known as the Rational Root Theorem. This theorem states that for a polynomial function f(x) = anxn + an-1xn-1 + ... + a1x + a0, where all coefficients are integers, any rational zero, expressed in its lowest terms as p/q, must have p as a factor of the constant term (a0) and q as a factor of the leading coefficient (an).
The steps to find all rational zeros are as follows:
- List all the factors of the constant term (a0) and call this list P.
- List all the factors of the leading coefficient (an) and call this list Q.
- Form a list of all possible ratios p/q, where p is from list P and q is from list Q. Be sure to include both positive and negative possibilities.
- Test each p/q by substituting these values into the polynomial function and check if it equals zero. This can be done using direct substitution or synthetic division. If the result is zero, then p/q is a rational zero.
- Once a rational zero is found, use polynomial long division or synthetic division to divide the original polynomial by (x - p/q) to simplify the polynomial.
- Repeat the process with the simplified polynomial until all zeros are found or you are left with an irreducible quadratic (or higher order) polynomial, which you can solve using other methods such as the quadratic formula.
Remember that not all polynomials have rational zeros, so there might be cases where this process yields no rational zeros. It's also important to note that other methods such as factoring, graphing, or using numerical methods might be needed to find other types of zeros.