Find all the real zeros of the polynomial. Use the quadratic formula if necessary. (Enter your answers as a comma-separated list. Enter all answers induding repetitions.) \[ P(x)=3 x^{3}+18 x^{2}+21 x-6 \] \( x= \) Need Help? Readit Watch :

To find the real zeros of the polynomial \( P(x) = 3x^3 + 18x^2 + 21x - 6 \), we can first look for rational roots using the Rational Root Theorem. Testing values, we find that \( x = -1 \) is a root. We can then divide the polynomial by \( (x + 1) \) to find the quadratic factor: Performing synthetic division, we get \( P(x) = (x + 1)(3x^2 + 15x - 6) \). Now, we can apply the quadratic formula to \( 3x^2 + 15x - 6 = 0 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-15 \pm \sqrt{15^2 - 4(3)(-6)}}{2(3)} = \frac{-15 \pm \sqrt{225 + 72}}{6} = \frac{-15 \pm \sqrt{297}}{6} \] Thus, the real zeros of the polynomial are: \[ x = -1, \frac{-15 + \sqrt{297}}{6}, \frac{-15 - \sqrt{297}}{6} \] Final answer: \( x = -1, \frac{-15 + \sqrt{297}}{6}, \frac{-15 - \sqrt{297}}{6} \) Real-life problem solvers often steer towards polynomial equations like this one when grappling with real-life occurrences, such as calculating dimensions in architecture that follow cubic relationships. Understanding how to efficiently strategize for roots can lead to more optimal solutions in critical scenarios. Additionally, it’s common to misjudge the polynomial's degree or overlook synthetic division; always ensure each term’s degree aligns when simplifying. Keeping your calculations clean and checking each step can save you from multiplying errors or missing potential roots!