Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. Enter all answers including repetitions.) \[ P(x)=3 x^{3}+17 x^{2}+21 x-9 \] Write the polynomial in factored form. \( P(x)=\square \)

To find the rational zeros of the polynomial \( P(x) = 3x^3 + 17x^2 + 21x - 9 \), we can use the Rational Root Theorem, which suggests that any rational solution will be a combination of factors of the constant term (-9) and the leading coefficient (3). The potential rational roots are: \[ \pm 1, \pm 3, \pm 9, \pm \frac{1}{3}, \pm \frac{3}{3}, \pm \frac{9}{3} \] This simplifies to: \[ \pm 1, \pm 3, \pm 9, \pm \frac{1}{3} \] Testing these values, we find that \( x = 1 \) is a root. Using synthetic division with \( x - 1 \), we divide the polynomial by this root: \[ P(x) = (x - 1)(3x^2 + 20x + 9) \] Next, we find the zeros of the quadratic \( 3x^2 + 20x + 9 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-20 \pm \sqrt{400 - 108}}{6} = \frac{-20 \pm \sqrt{292}}{6} = \frac{-20 \pm 2\sqrt{73}}{6} = \frac{-10 \pm \sqrt{73}}{3} \] Thus, the rational roots are simply \( x = 1 \) with multiplicity, and the additional zeros are irrational. In total: The rational zeros of \( P(x) \): \( 1 \), \( 1 \), \( 1 \) The polynomial in factored form is: \[ P(x) = (x - 1)^3(3x^2 + 20x + 9) \] Now let's provide some engaging knowledge! The discovery of rational roots can feel like a treasure hunt for mathematicians! By using clever techniques like synthetic division, you can simplify complex polynomials into manageable factors. This method transforms what seems daunting into a series of straightforward steps—almost like following a recipe in the kitchen! Once you find those elusive roots, it's time to highlight their importance. Understanding rational roots doesn't just help in solving equations; it also plays a critical role in fields such as engineering and physics, where polynomial equations frequently arise. Whether it's calculating trajectories or modeling dynamic systems, rational roots ensure you have accurate solutions that can lead to innovative designs!