Questions: Determine the remaining solutions of the cubic equation: Find all solutions to (x^3-2 x^2-10 x+8=0) if (x=4) is one solution. The solutions are (x=4), and (pm sqrt )

Solution Steps

To find the remaining solutions of the cubic equation given that \( x = 4 \) is one solution, we can use polynomial division to divide the cubic equation by \( x - 4 \). This will give us a quadratic equation. We can then solve the quadratic equation using the quadratic formula to find the other two solutions.

We are given the cubic equation \( x^{3} - 2x^{2} - 10x + 8 = 0 \) and know that \( x = 4 \) is one of its solutions.

To find the remaining solutions, we divide the cubic polynomial by \( x - 4 \). This results in a quadratic polynomial, which we can express as: \[ x^{2} + 2x - 2 = 0 \]

We apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \) to find the roots of the quadratic equation. Here, \( a = 1 \), \( b = 2 \), and \( c = -2 \). The discriminant is calculated as: \[ b^{2} - 4ac = 2^{2} - 4(1)(-2) = 4 + 8 = 12 \] Thus, the roots are: \[ x = \frac{-2 \pm \sqrt{12}}{2} = \frac{-2 \pm 2\sqrt{3}}{2} = -1 \pm \sqrt{3} \]

The complete set of solutions to the original cubic equation is: \[ x = 4, \quad x = -1 + \sqrt{3}, \quad x = -1 - \sqrt{3} \]

Final Answer