Answer:
x = 1, x = 7, and x = -1.
Explanation:
o solve the polynomial equation x^3 - 7x^2 - x + 7 = 0, we can use a combination of synthetic division and factoring by grouping:
First, we need to find a root of the polynomial using the Rational Root Theorem. The possible rational roots of the polynomial are the factors of 7 (the constant term) divided by the factors of 1 (the leading coefficient), or ±1, ±7. By trying each of these values in the polynomial, we find that x = 1 is a root.
Using synthetic division, we can divide the polynomial by (x - 1) to obtain a quadratic equation:
1 | 1 -7 -1 7
| 1 -6 -7
|_____________
1 -6 -7 0
Therefore, (x - 1) is a factor of the polynomial, and we have:
x^3 - 7x^2 - x + 7 = (x - 1)(x^2 - 6x - 7)
Now we need to solve the quadratic equation x^2 - 6x - 7 = 0. We can factor it as (x - 7)(x + 1) = 0, so the solutions are x = 7 and x = -1.
Therefore, the solutions to the original polynomial equation x^3 - 7x^2 - x + 7 = 0 are x = 1, x = 7, and x = -1.