3. Solve for \( x \) for the following: \( \quad(7 \) marks) \( 2 x^{3}+11 x^{2}-60 x+27=0 \)

To solve the equation \( 2x^3 + 11x^2 - 60x + 27 = 0 \), we can start by using synthetic division or the Rational Root Theorem to find potential rational roots. Testing small integer values, we find \( x = 3 \) works because: \[ 2(3^3) + 11(3^2) - 60(3) + 27 = 54 + 99 - 180 + 27 = 0 \] Now that we have a root, we can perform synthetic division with \( x - 3 \) to factor the polynomial further: ``` 3 | 2 11 -60 27 | 6 51 -27 --------------------- 2 17 -9 0 ``` This gives us \( 2x^2 + 17x - 9 = 0 \). We can now use the quadratic formula to find the remaining roots: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-17 \pm \sqrt{17^2 - 4(2)(-9)}}{2(2)} = \frac{-17 \pm \sqrt{289 + 72}}{4} = \frac{-17 \pm \sqrt{361}}{4} \] Calculating \( \sqrt{361} = 19 \): \[ x = \frac{-17 \pm 19}{4} \] This gives us two solutions: 1. \( x = \frac{2}{4} = \frac{1}{2} \) 2. \( x = \frac{-36}{4} = -9 \) Thus, the solutions for \( x \) are: \[ x = 3, \quad x = \frac{1}{2}, \quad x = -9 \]