Final answer:
To find the rational zeros of the polynomial 2x⁴ - 15x³ + 35x² - 30x + 8, use the Rational Root Theorem to list possible rational zeros and test each by synthetic division or substitution into the polynomial.
Step-by-step explanation:
The student has asked for the rational zeros of the polynomial 2x⁴ - 15x³ + 35x² - 30x + 8. To find the rational zeros of a polynomial, we can use the Rational Root Theorem, which states that any rational solution, expressed as a fraction p/q, will have p as a factor of the constant term of the polynomial and q as a factor of the leading coefficient. In this case, the constant term is 8, and the leading coefficient is 2. So the possible values for p are ±1, ±2, ±4, ±8, and the possible values for q are ±1, ±2. Therefore, the possible rational zeros are ±1, -1, ±2, -2, ±4, -4, ±1/2, -1/2, ±2/2, -2/2, ±4/2, -4/2, ±8/2, and -8/2.
Once the potential rational zeros are determined, each can be tested using synthetic division or by substituting them into the polynomial to see if they yield a zero value. If the remainder is zero, then the tested candidate is indeed a zero of the polynomial. More advanced methods like the use of graphing technology or factoring techniques can further aid in finding zeros for higher degree polynomials that cannot be easily factored.