Final answer:
The student's question involves finding the roots of a quartic polynomial, which can involve methods such as factorization, synthetic division, and the quadratic formula. Rational Root Theorem may also help in identifying possible rational roots to test and reduce the polynomial to a lower degree.
Step-by-step explanation:
The student is asking for the roots of the polynomial function f(x) = x⁴ - 10x³ + 42x² - 88x + 80. To find the roots of this quartic polynomial, one method is to factor by grouping or to use synthetic division if one can guess a root. It's often helpful to check if the polynomial has any rational roots using the Rational Root Theorem, which tells us that the possible rational roots are the factors of the constant term divided by the factors of the leading coefficient. In this case, the factors of 80 (the constant term) are ± 1, ± 2, ± 4, ± 5, ± 8, ± 10, ± 16, ± 20, ± 40, ± 80. Since the leading coefficient is 1, we can just consider these values as potential roots.
If we were to find one root, we could then perform synthetic division or polynomial long division to reduce the quartic to a cubic, and then to a quadratic if another root is found, at which point we can use the quadratic formula to find any remaining roots. Alternatively, numerical methods or computer algebra systems can be used to approximate the roots or to factor the polynomial when factorization is not obvious.