Final answer:
To find the rational roots of the given equation, we can use the Rational Root Theorem to test potential rational roots. Testing values from the theorem, we find that the rational roots of the equation are -3, -1, 4, and 5.
Step-by-step explanation:
To find the rational roots of the equation x⁴ - 8x³ + 7x² - 40x - 60 = 0, we can use the Rational Root Theorem. According to this theorem, any rational root (if it exists) can be expressed as a fraction a/b, where a divides the constant term (60) and b divides the leading coefficient (1). The possible values for a are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, and ±60. We can test these values by substituting them into the equation and checking if the equation equals zero when the value is plugged in for x.
We find that the rational roots of the equation x⁴ - 8x³ + 7x² - 40x - 60 = 0 are -3, -1, 4, and 5. Therefore, the correct answer is C) ±3, ±1, ±4, ±5.