The function below has at least one rational zero. Use this fact to find all zeros of the function. \[ f(x)=2 x^{3}+9 x^{2}+5 x-7 \] If there is more than one zero, separate them with commas. Winite evact values, not decimal approximations. \( \square \)

To find all zeros of the function \( f(x) = 2x^3 + 9x^2 + 5x - 7 \), we start by identifying the rational zeros using the Rational Root Theorem. ### Step 1: Possible Rational Zeros The possible rational zeros are of the form \(\frac{p}{q}\), where \( p \) is a factor of the constant term \(-7\) and \( q \) is a factor of the leading coefficient \(2\). Thus, the possible rational zeros are: \[ \pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2} \] ### Step 2: Testing the Possible Zeros We substitute these values into the function to find which ones are actual zeros. - **Testing \( x = -\frac{7}{2} \):** \[ f\left(-\frac{7}{2}\right) = 2\left(-\frac{7}{2}\right)^3 + 9\left(-\frac{7}{2}\right)^2 + 5\left(-\frac{7}{2}\right) - 7 = 0 \] So, \( x = -\frac{7}{2} \) is a zero. ### Step 3: Polynomial Division We factor out \( \left(x + \frac{7}{2}\right) \) from \( f(x) \) using synthetic division: \[ f(x) = \left(x + \frac{7}{2}\right)(2x^2 + 2x - 2) \] ### Step 4: Solving the Quadratic Factor Next, we solve the quadratic equation \( 2x^2 + 2x - 2 = 0 \): \[ x^2 + x - 1 = 0 \] Using the quadratic formula: \[ x = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \] ### Final Zeros Thus, all zeros of the function \( f(x) \) are: \[ -\frac{7}{2}, \quad \frac{-1 + \sqrt{5}}{2}, \quad \frac{-1 - \sqrt{5}}{2} \] **Answer:** $-\dfrac{7}{2},\ \dfrac{-1+\sqrt{5}}{2},\ \dfrac{-1-\sqrt{5}}{2}$