6х³-18х²-144х+96=0

To solve the cubic equation \(6x^3 - 18x^2 - 144x + 96 = 0\), we can begin by simplifying the equation. First, let's factor out the greatest common factor (GCF) from each term: \[ 6(x^3 - 3x^2 - 24x + 16) = 0 \] We can divide both sides by 6 (since 6 is not equal to zero): \[ x^3 - 3x^2 - 24x + 16 = 0 \] Next, we can look for rational roots using the Rational Root Theorem. The possible rational roots are the factors of the constant term (16) divided by the leading coefficient (1). Thus, potential rational roots are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16\). Let's test these possible roots: 1. Test \(x = 2\): \[ 2^3 - 3(2^2) - 24(2) + 16 = 8 - 12 - 48 + 16 = -36 \quad \text{(not a root)} \] 2. Test \(x = 4\): \[ 4^3 - 3(4^2) - 24(4) + 16 = 64 - 48 - 96 + 16 = -64 \quad \text{(not a root)} \] 3. Test \(x = -2\): \[ (-2)^3 - 3(-2)^2 - 24(-2) + 16 = -8 - 12 + 48 + 16 = 44 \quad \text{(not a root)} \] 4. Test \(x = -4\): \[ (-4)^3 - 3(-4)^2 - 24(-4) + 16 = -64 - 48 + 96 + 16 = 0 \quad \text{(is a root)} \] Since \(x = -4\) is a root, we can perform polynomial long division to divide \(x^3 - 3x^2 - 24x + 16\) by \(x + 4\): Perform the division: \[ x^3 - 3x^2 - 24x + 16 \div (x + 4) \] The result will be: \[ x^2 - 7x + 4 \] Now, we can factor this quadratic equation: To find the roots of \(x^2 - 7x + 4 = 0\), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 1\), \(b = -7\), and \(c = 4\): \[ x = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(4)}}{2(1)} \] \[ x = \frac{7 \pm \sqrt{49 - 16}}{2} \] \[ x = \frac{7 \pm \sqrt{33}}{2} \] Thus, the solutions to the original equation \(6x^3 - 18x^2 - 144x + 96 = 0\) are: 1. \(x = -4\) 2. \(x = \frac{7 + \sqrt{33}}{2}\) 3. \(x = \frac{7 - \sqrt{33}}{2}\) These are the three roots of the cubic equation.