Final answer:
To factor the polynomial function p(x) = x^4 + x^3 - x^2 + x - 2, use the known x-intercept at x = -2 to determine that (x + 2) is a factor and divide p(x) by (x + 2) to find the remaining factors.
Step-by-step explanation:
The polynomial function p(x) = x4 + x3 - x2 + x - 2 has a given x-intercept at x = -2. To factor p(x) completely, we begin by using this x-intercept to find its corresponding factor of the polynomial.
If p(x) has an x-intercept at x = -2, this means p(-2) = 0. Therefore, x + 2 is a factor of p(x). We can perform polynomial division or use synthetic division to divide p(x) by x + 2 to find the other factors.
After division, the polynomial reduces to q(x) = x3 - x2 - 2x + 1. We can now attempt to factor this polynomial, looking for rational roots or simpler factors.
If further factoring by inspection is difficult, we can use techniques such as the Rational Root Theorem or synthetic division to find the remaining factors of q(x), yielding the complete factorization of p(x).