A polynomial \( P \) is given. \[ P(x)=x^{3}-6 x^{2}-9 x+54 \] (a) Find all the real zeros of \( P \). (Enter your answers as a comma-separated list. Enter all answers including repetitions.) \( x= \) (b) Sketch the graph of \( P \).

To find the real zeros of the polynomial \( P(x) = x^{3} - 6x^{2} - 9x + 54 \), we can use the Rational Root Theorem or synthetic division to test potential rational roots. After trying several values, we can find that \( x = 3 \) is a root. By performing polynomial division or synthetic division of \( P(x) \) by \( (x - 3) \), we can factor the polynomial to find the other zeros: \( P(x) = (x - 3)(x^2 - 3x - 18) \). Next, we can solve \( x^2 - 3x - 18 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This leads to: 1. \( x = \frac{3 + \sqrt{81}}{2} = 8 \) 2. \( x = \frac{3 - \sqrt{81}}{2} = -2 \) Thus, the real zeros of \( P \) are \( x = 3, 8, -2 \). Now, to sketch the graph of \( P \), we note some key features: 1. The degree of the polynomial is odd, indicating that the ends of the graph will go in opposite directions (left will go down, right will go up). 2. The zeros we found will be the x-intercepts at \( x = 3, 8, -2 \). 3. The polynomial will reach a local maximum and minimum between these zeros, so we can evaluate \( P \) at a few points (e.g., \( P(0) = 54 \), positive indicating the graph starts high). 4. Sketch the curve, starting high (from the left), crossing \( (−2, 0) \), ascending to a peak before crossing at \( (3, 0) \), going down, then crossing again at \( (8, 0) \) and ending high (to the right). In summary, the real zeros of \( P \) are: \( x = -2, 3, 8 \). --- Polynomials have a rich history, dating back to ancient civilizations such as the Babylonians and Greeks, who used them for their mathematical principles in trade and astronomy. As math evolved, so did polynomials, becoming a vital tool in various fields—laying the groundwork for calculus and modern algebra! When dealing with polynomial roots, a common mistake is failing to check for repeated roots or extraneous solutions. Always perform a thorough verification of potential roots and consider using synthetic division or factoring to simplify polynomials correctly; it can save time and avert errors in calculations!