which of the following is the gcf of the polynomial 4x^4+12x^3−36x^2+24x?

Answer 1

To find the greatest common factor (GCF) of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\), we will start by identifying the GCF of the coefficients and the variable factors in each term.

1. Find the GCF of the coefficients (4, 12, -36, 24):

   • The factors of 4 are: \(1, 2, 4\)
   • The factors of 12 are: \(1, 2, 3, 4, 6, 12\)
   • The factors of -36 are: \(1, 2, 3, 4, 6, 9, 12, 18, 36\)
   • The factors of 24 are: \(1, 2, 3, 4, 6, 8, 12, 24\)

   The common factors of all coefficients are \(1, 2, 4\). The greatest of these is \(4\).

2. Find the GCF of the variable factors:

   • The terms in the polynomial are \(4x^4\), \(12x^3\), \(-36x^2\), and \(24x\).
   • The term with the lowest power of \(x\) is \(x\), which appears in all terms.

3. Combine the GCFs:

   • The GCF of the coefficients is \(4\) and the lowest power of \(x\) is \(x^1\).

Thus, the GCF of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\) is:

\[ \text{GCF} = 4x \]

So the answer is \(4x\).