What is the factored form of 6n^4 – 24n^3 + 18n?

To find the factored form of the expression 6n⁴ – 24n³ + 18n, we begin by looking for common factors in all the terms.

First, we can factor out the greatest common factor (GCF). The GCF of the coefficients (6, -24, and 18) is 6, and the lowest power of n in the terms is n (since all terms are divisible by n6n:

6n(n³ – 4n² + 3)

Now we need to factor the expression inside the parentheses. We want to factor the cubic polynomial n³ – 4n² + 3.

To factor this further, we can try to find possible rational roots using the Rational Root Theorem. Testing some values, we find that n = 1 is a root:

1³ – 4(1)² + 3 = 1 – 4 + 3 = 0

Now, we can divide the cubic polynomial by (n – 1) using synthetic division, which gives us:

n³ – 4n² + 3 = (n – 1)(n² – 3n – 3)

Next, we can factor the quadratic n² – 3n – 3. We are looking for two numbers that multiply to -3 and add to -3. Unfortunately, this quadratic cannot be factored further with rational coefficients.