Final answer:
Using synthetic division and factoring, the zeros of the polynomial function f(x) = x^3 + 12x^2 + 41x + 30 are found to be -5, -6, and -1. The correct answer from the given options is B: {-5, -6, -1}.
Step-by-step explanation:
To use synthetic division to divide f(x) = x^3 + 12x^2 + 41x + 30 by x + 5, we first need to identify the zero of the divisor, which in this case is -5. Set up the synthetic division as follows:
- Write down the coefficients of f(x): 1, 12, 41, 30.
- Write the zero of the divisor outside the division symbol: -5.
- Carry down the leading coefficient: 1.
- Multiply -5 by 1 and write the result under the next coefficient: -5 * 1 = -5.
- Add the second coefficient and the result: 12 + (-5) = 7.
- Continue this process until you have processed all coefficients.
The synthetic division will look like this:
-5 | 1 12 41 30
| -5 -35 -30
-----------------
1 7 6 0
The result gives us a quotient of x^2 + 7x + 6 and a remainder of 0, indicating that -5 is a zero of f(x). Factor the quadratic to find the other zeros:
x^2 + 7x + 6 = (x + 6)(x + 1)
Therefore, the zeros are -5, -6, and -1. The correct answer is B: {-5, -6, -1}.