If f(–5) = 0, what are all the factors of the function f(x) = x³ - 19x + 30? Use the remainder theorem.

The remainder theorem is stated as follows: When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k).

Answer: The factors of the function f(x)= x³ - 19x + 30 are (x + 5), (x - 2), and (x - 3).

Let's understand the solution in detail.

Explanation:

Given function: f(x) = x³ - 19x + 30.

Since f(-5) = 0, hence, by remainder theorem, (x + 5) = 0.

Hence, (x + 5) is a factor of f(x) = x³ - 19x + 30.

Now, after dividing f(x) = x³ - 19x + 30 by (x + 5), we get x² - 5x + 6.

So, f(x) = (x + 5) (x² - 5x + 6).

Now, we find the solutions for (x² - 5x + 6) by splitting the middle term.

⇒ x² - 5x + 6

⇒ x² - 3x - 2x + 6

⇒ x(x - 3) - 2(x - 3)

⇒ (x - 2) (x - 3)

Now, we see that f(x) = (x + 5)(x - 2)(x - 3).

Hence, the factors of the function f(x) = x³ - 19x + 30 is (x + 5), (x - 2), and (x - 3).