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

Solution:

It is given that

f(x) = x³ - 19x + 30

If f(-5) = 0, it means that (x + 5) is a factor of f(x)

We have to find the other factors

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).

Using the remainder theorem we get

f(x) = (x + 5) (x² + 5x + 6)

By splitting the middle term

f(x) = (x + 5) (x² - 2x - 3x + 6)

Taking out the common terms

f(x) = (x + 5) [x (x - 2) - 3 (x - 2)]

f(x) = (x + 5) (x - 2) (x - 3)

Therefore, all the factors of the function are (x + 5) (x - 2) (x - 3).

---

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

Summary:

If(-5) = 0, all the factors of the function f(x) = x³ - 19x + 30 are (x + 5) (x - 2) (x - 3).