To find the solutions for the cubic equation x^3 - 7x + 6 = 0, you can use various methods, such as the Rational Root Theorem, synthetic division, or numerical methods like Newton-Raphson. Here, I'll use the Rational Root Theorem to find potential rational solutions and then use synthetic division to confirm them:
The Rational Root Theorem states that if a rational number p/q is a root of the polynomial equation, then p is a factor of the constant term (here, 6), and q is a factor of the leading coefficient (here, 1).
The factors of 6 are ±1, ±2, ±3, and ±6.
The factors of 1 are ±1.
Now, we'll try these potential rational solutions one by one:
1. x = 1
- Plug x = 1 into the equation: (1^3) - 7(1) + 6 = 1 - 7 + 6 = 0.
- x = 1 is a root of the equation.
Now that we have one solution, we can use synthetic division to find the other solutions:
Perform synthetic division with (x - 1) to factor out (x - 1) from the equation:
```
1 | 1 0 -7 6
| - 1 -6
|_______________
1 1 -6 0
```
The resulting quadratic equation is x^2 + x - 6 = 0, which can be factored as (x + 3)(x - 2).
2. Using x + 3 = 0, we get x = -3.
3. Using x - 2 = 0, we get x = 2.
So, the three solutions to the cubic equation x^3 - 7x + 6 = 0 are:
1. x = 1
2. x = -3
3. x = 2.