Final answer:
To find the Rational Zeros of the function f(x) = 12x^3 + 73x^2 + 5x - 6, we use the Rational Zero Theorem to identify possible candidates and then test them to find actual zeros. After finding a zero, the polynomial can be factored to find the remaining zeros.
Step-by-step explanation:
To find the Rational Zeros of the polynomial function f(x) = 12x^3 + 73x^2 + 5x - 6, we use the Rational Zero Theorem. This theorem states that any rational zero, expressed in its lowest terms p/q, is such that p is a factor of the constant term (in this case -6) and q is a factor of the leading coefficient (in this case 12).
Therefore, the possible rational zeros are ±1, ±2, ±3, ±6, ±1/2, ±1/3, ±1/4, ±1/6, ±2/3, and ±1/12. We can test these numbers using synthetic division or by plugging them into the function to see if they yield a value of 0.
Once we find a rational zero, we can then factor the polynomial either by synthetic division or by factoring by grouping. This will reduce the polynomial to a second degree, which we can then solve by either factoring further, if possible, or using the quadratic formula to find the remaining zeros.
The complete question is: Find the Rational Zeros and Factor: f(x)=12x^(3)+73x^(2)+5x-6 is: