Answer:The zeroes of f(x) = x³ + x² - 22x - 40 are x = -2, x = -4, and x = 5.
Step-by-step explanation: To find the zeroes of the function f(x) = x³ + x² - 22x - 40, we need to determine the values of x that make the function equal to zero.
One way to find the zeroes is by factoring the polynomial. However, factoring a cubic polynomial can be complex and may not always be possible. In this case, we can use the rational root theorem and synthetic division to find the zeroes.
According to the rational root theorem, if a rational number p/q is a zero of the polynomial, then p is a factor of the constant term (-40) and q is a factor of the leading coefficient (1).
By testing the factors of -40 (±1, ±2, ±4, ±5, ±8, ±10, ±20, ±40) as potential rational zeroes, we find that x = -2, x = -4, and x = 5 are the zeroes of the function.
