Answer:
All the zeroes of f(x) are x = 1, x = -1 + 4i and x = -1 - 4i.
Explanation:
Given that f(x) = x³ + x² + 15x - 17
Now, we have to find all the zeroes of the function.
Given that x = - 1 + 4i is a zero of the function.
So, x = - 1 - 4i must be another zero of the function.
Therefore, (x + 1 - 4i)(x + 1 + 4i) will be factor of the function.
Hence, (x + 1 - 4i)(x + 1 + 4i)
= x² + 2x + (1 - 4i)(1 + 4i)
= x² + 2x + [1² - (4i)²]
= x² + 2x + 17
Assume that (x + a) is another factor of f(x).
Therefore, we can write f(x) = x³ + x² + 15x - 17 = (x + a)(x² + 2x + 17)
⇒ x³ + x² + 15x - 17 = x³ + (a + 2)x² + (2a + 17)x + 17a
Hence, comparing the coefficients we can write
a + 2 = 1
⇒ a = -1
Therefore, f(x) =x³ + x² + 15x - 17 = (x - 1)(x² + 2x + 17)
So, all the zeroes of f(x) are x = 1, x = -1 + 4i and x = -1 - 4i (Answer)