Final answer:
To find the zeros of the function f(x) = 5x³ - 12x² - 1272x + 3, we use the Rational Root Theorem to identify potential rational zeros, test them, and then factorize the polynomial or apply the quadratic formula to find the exact values of all zeros.
Step-by-step explanation:
To find all zeros of the function f(x) = 5x³ - 12x² - 1272x + 3, we must first recognize that there might be rational zeros based on the Rational Root Theorem. This theorem states that any rational zero of a polynomial function, where all coefficients are integers, must be a fraction that has a numerator which is a factor of the constant term and a denominator that is a factor of the leading coefficient. For our function, the factors of the constant term (3) are ±1, and ±3, and the factors of the leading coefficient (5) are ±1, and ±5.
We test these potential rational zeros using synthetic division or the Remainder Theorem. Once we find a rational zero, we can use it to factorize the polynomial using polynomial division. If we cannot factorize the polynomial directly, we might use techniques such as completing the square or the quadratic formula to solve for the remaining zeros.
After finding all the zeros, we list them, separating by commas, and write their exact values, not decimal approximations. Remember that apart from rational zeros, a cubic function may also have irrational or complex zeros, so ensure all possibilities are considered when solving for zeros.