Use the rational zeros theorem to find all the real zeros of the polynomial function. Use the zeros to factor fover the real numbers. \( f(x)=2 x^{3}-4 x^{2}-34 x+68 \) Find the real zeros of \( f \). Select the correct choice below and, if necessary, fill in the answer box to complete your answer. A. \( x=\square \)

To find the real zeros of the polynomial function \( f(x) = 2x^3 - 4x^2 - 34x + 68 \) using the Rational Root Theorem, we first identify the possible rational roots. The Rational Root Theorem states that any rational solution, in the form of \( \frac{p}{q} \), has \( p \) as a factor of the constant term and \( q \) as a factor of the leading coefficient. 1. **Identify the constant term and leading coefficient:** - Constant term: \( 68 \) - Leading coefficient: \( 2 \) 2. **Find the factors of the constant term (68):** - Factors of \( 68 \): \( \pm 1, \pm 2, \pm 4, \pm 17, \pm 34, \pm 68 \) 3. **Find the factors of the leading coefficient (2):** - Factors of \( 2 \): \( \pm 1, \pm 2 \) 4. **List the possible rational roots:** - Possible rational roots: \( \pm 1, \pm 2, \pm 4, \pm 17, \pm 34, \pm 68 \) divided by \( \pm 1 \) and \( \pm 2 \): - This gives us the possible rational roots: \( \pm 1, \pm 2, \pm 4, \pm 17, \pm 34, \pm 68, \pm \frac{1}{2}, \pm \frac{17}{2} \) 5. **Test the possible rational roots:** We can test these values in the polynomial \( f(x) \). - Testing \( x = 2 \): \[ f(2) = 2(2)^3 - 4(2)^2 - 34(2) + 68 = 2(8) - 4(4) - 68 + 68 = 16 - 16 - 68 + 68 = 0 \] So, \( x = 2 \) is a root. 6. **Perform synthetic division to factor the polynomial:** We can divide \( f(x) \) by \( (x - 2) \) using synthetic division. \[ \begin{array}{r|rrrr} 2 & 2 & -4 & -34 & 68 \\ & & 4 & 0 & -68 \\ \hline & 2 & 0 & -34 & 0 \\ \end{array} \] The result is \( 2x^2 - 34 \). 7. **Factor the quadratic:** Now we can factor \( 2x^2 - 34 \): \[ 2x^2 - 34 = 2(x^2 - 17) = 2(x - \sqrt{17})(x + \sqrt{17}) \] 8. **Combine the factors:** Thus, the complete factorization of \( f(x) \) over the real numbers is: \[ f(x) = 2(x - 2)(x - \sqrt{17})(x + \sqrt{17}) \] 9. **Find all real zeros:** The real zeros of \( f(x) \) are: - \( x = 2 \) - \( x = \sqrt{17} \) - \( x = -\sqrt{17} \) So, the real zeros of \( f \) are: - \( x = 2 \) - \( x = \sqrt{17} \) - \( x = -\sqrt{17} \) ### Final Answer: A. \( x = 2 \) (and also \( x = \sqrt{17} \) and \( x = -\sqrt{17} \) as additional zeros).