Explanation:
We are asked to solve the equation x³ – 7x + 6 = 0, which is a cubic equation. We will find the roots using the trial-and-error method followed by factorization.
1. Check for a Root via Substitution: We can start by trying simple integer values of x to check for roots. Try x = 1:
f (1) = 1³ – 7(1) + 6 = 1 – 7 + 6 = 0
Therefore, x = 1 is a root of the polynomial.
2. Perform Synthetic Division: Now, we divide x³ – 7x + 6 by x  – 1 using synthetic division:
The quotient is x² + x  – 6, and the remainder is 0.
3. Factor the Resulting Quadratic: Now, factor the quadratic x² + x – 6. We look for two numbers that multiply to –6 and add up to 1. These numbers are 3 and –2.
So, the quadratic factors as:
                                           x² + x – 6 = (x – 2)(x + 3)
4. Final Factorization: Therefore, the factorization of the cubic equation is:
(x – 1)(x – 2)(x + 3) = 0
The solutions are:
x = 1, x = 2, x = –3