Solve the cubic equation, x³ − 7x² + 14x − 8 = 0 and Find all real roots.

We first try to find at least one rational root. By Rational Root Theorem, possible rational roots are factors of 8: ±1, ±2, ±4, ±8 Test x = 1: 1³ − 7(1²) + 14(1) − 8 = 1 − 7 + 14 − 8 = 0 ✅ So (x − 1) is a factor. Now divide the cubic by (x − 1) using synthetic division: Coefficients: 1, −7, 14, −8 Bring down 1 Multiply 1×1 = 1, add to −7 → −6 Multiply 1×(−6) = −6, add to 14 → 8 Multiply 1×8 = 8, add to −8 → 0 So the cubic becomes: (x − 1)(x² − 6x + 8) = 0 Now solve the quadratic: x² − 6x + 8 = 0 Factor: (x − 2)(x − 4) = 0 So the full factorization is: (x − 1)(x − 2)(x − 4) = 0 Therefore, x = 1, 2, 4