What is the factored form of 6n⁴ - 24n³ + 18n?

Solution:

Let equation f(n) = 6n⁴ - 24n³ + 18n

f(n) = 6n[n³ - 4n² + 3]

Take F(n) = n³ - 4n² + 3

F(1) = (1)³ - 4(1)² + 3

F(1) = 0

(n - 1) is one of the factors of F(n).

By synthetic division

1 │ 1  -4  0  3
   │ 0  1  -3  -3
--------------------
      1  -3  -3  0

Quotient = n² - 3n -3

The discriminant = b² - 4ac

= (-3)² - [4(1)(-3)]

= 21

⇒ n² - 3n -3 cannot be factored further

Therefore, the factors of f(n) = 6n(n -1)(n² - 3n -3)

---

What is the factored form of 6n⁴ - 24n³ + 18n?

Summary:

The factored form of 6n⁴ - 24n³ + 18n is 6n(n -1)(n² - 3n -3).