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

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Finding the Factored Form of 6n4 – 24n3 + 18n

To factor the polynomial 6n4 – 24n3 + 18n, we can follow these steps:

  1. Identify the greatest common factor (GCF):

    2. The first step in factoring any polynomial is to find the GCF of all the terms. The terms of our polynomial are:

    • 6n4
    • -24n3
    • 18n

    The GCF of these terms is 6n, as 6 is the largest coefficient that divides all the numerical coefficients (6, -24, and 18), and n is the lowest power of n present in all terms (which is n).

  2. Factor out the GCF:

    3. Now that we have the GCF, we can factor it out:

    6n (n3 - 4n2 + 3)
  3. Factor the remaining polynomial:

    4. Next, we will factor the polynomial n3 – 4n2 + 3. To do this, we can look for possible rational roots using the Rational Root Theorem or apply other factoring techniques. Testing possible roots, we find:

    • Testing n = 1:
      13 – 4(12) + 3 = 0 (So, n-1 is a factor)
    • Therefore, we can perform synthetic division or polynomial long division to factor further.

    After performing synthetic division, we can express:

    n3 - 4n2 + 3 = (n - 1)(n2 - 3n - 3)

    The quadratic n2 – 3n – 3 can also be factored or solved using the quadratic formula. However, it does not factor nicely into integers. So we will leave it as is.

Final Factored Form

Putting it all together, the full factored form of the original polynomial 6n4 – 24n3 + 18n is:

6n(n - 1)(n2 - 3n - 3)

This represents the polynomial completely factored over the integers. In conclusion, the factored form is:

6n(n – 1)(n2 – 3n – 3)

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