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

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

The Correct Answer and Explanation is:

To factor the expression (6n^4 – 24n^3 + 18n), we start by looking for common factors in all the terms.

1. Identify the Greatest Common Factor (GCF):
   The coefficients of the terms are (6), (-24), and (18). The GCF of these numbers is (6).
   The variable parts are (n^4), (n^3), and (n). The smallest power of (n) is (n). Therefore, the GCF of the entire expression is (6n).
2. Factor out the GCF:
   We divide each term by the GCF (6n):
   [
   6n^4 \div 6n = n^3,
   ]
   [
   -24n^3 \div 6n = -4n^2,
   ]
   [
   18n \div 6n = 3.
   ]
   Thus, factoring out (6n) from the original expression gives:
   [
   6n(n^3 – 4n^2 + 3).
   ]
3. Factor the cubic expression (n^3 – 4n^2 + 3):
   We need to factor the polynomial (n^3 – 4n^2 + 3). We can attempt to find rational roots using the Rational Root Theorem, which suggests testing factors of the constant term ((3)) over the leading coefficient ((1)). The possible rational roots are (\pm 1, \pm 3). Testing (n = 1):
   [
   1^3 – 4(1^2) + 3 = 1 – 4 + 3 = 0.
   ]
   Since (n = 1) is a root, we can factor (n^3 – 4n^2 + 3) using synthetic division by (n – 1): [
   \begin{array}{r|rrr}
   1 & 1 & -4 & 0 & 3 \
   & & 1 & -3 & -3 \
   \hline
   & 1 & -3 & -3 & 0 \
   \end{array}
   ]
   This yields (n^2 – 3n – 3).
4. Final Factored Form:
   Thus, we can express the original polynomial as:
   [
   6n(n – 1)(n^2 – 3n – 3).
   ]
   To factor (n^2 – 3n – 3) further, we can apply the quadratic formula if necessary, but for now, we keep it in this form. The fully factored expression is:
   [
   6n(n – 1)(n^2 – 3n – 3).
   ]

In summary, the factored form of (6n^4 – 24n^3 + 18n) is (6n(n – 1)(n^2 – 3n – 3)). This process demonstrates the steps of identifying the GCF, performing polynomial division, and recognizing roots to achieve the complete factorization.