We are tasked with factorizing the polynomial $x^4 + x^3 - 7x^2 - x + 6$ as a product of four linear factors.

Steps to Factorize:

1. Use the Rational Root Theorem to identify potential roots (factors). The possible rational roots are the factors of the constant term (+6) divided by the factors of the leading coefficient (which is 1). The possible rational roots are: $\pm 1, \pm 2, \pm 3, \pm 6$

2. Test these values in the polynomial using synthetic or direct substitution to check if any of them are roots. If a value is a root, then $x - \text{(that value)}$ is a factor of the polynomial.

3. Perform Polynomial Division once a root is found to reduce the degree of the polynomial, and continue factorizing the remaining polynomial.

Let me first try some of the possible roots to find the factors. I'll proceed with the calculations.The roots of the polynomial $x^4 + x^3 - 7x^2 - x + 6$ are $-3$, $-1$, $1$, and $2$. Therefore, the polynomial can be factorized as:

$(x + 3)(x + 1)(x - 1)(x - 2)$

Thus, the factorization of the polynomial into four linear factors is:

$x^4 + x^3 - 7x^2 - x + 6 = (x + 3)(x + 1)(x - 1)(x - 2)$

Would you like more details on any of the steps or have further questions?

Related questions:

1. How does the Rational Root Theorem work in detail?
2. Can you explain the process of synthetic division for this polynomial?
3. What happens if the polynomial had complex roots instead of real ones?
4. How can I factor polynomials that don't have rational roots?
5. What is the next step after finding the roots to verify the factorization?

Tip: Always verify your factorization by expanding the factors back into the original polynomial to ensure correctness.