Final answer:
The polynomial with zeros of 13, 5i, and −5i, and a value of −680 at x=3 is 2x³ - 26x² + 50x - 650 when expanded, but this doesn't match any of the given answer choices.
Step-by-step explanation:
To find a third-degree polynomial with zeros of 13, 5i, and −5i, and a value of −680 when x=3, we start by using the fact that the zeros of the polynomial can be used to create factors of the polynomial. Since the polynomial has real coefficients, the complex zeros must come in conjugate pairs, which we have with 5i and −5i. We can form two factors from these zeros: (x - 13) from the real zero 13, and (x - 5i)(x + 5i) = x² + 25 from the complex zeros.
The polynomial in factored form is k(x - 13)(x² + 25), where k is a constant. To determine the value of k, we use the given value of the polynomial at x=3:
f(3) = k(3 - 13)((3)² + 25) = k(-10)(9 + 25) = -680 k(-10)(34) = -680 k = −680 / (-10)(34) k = 2
Therefore, the polynomial in factored form with k=2 is:
f(x) = 2(x - 13)(x² + 25)
Expanding this, we find the polynomial in standard form:
f(x) = 2x³ - 26x² + 50x - 650
However, none of the choices given in the original question (a-d) are correct. There might be a mistake in the given options or the way the question was transcribed. It is important to double-check the provided choices or confirm with additional details for the correct answer