To find the greatest common factor (GCF) of the polynomial \( 4x^4 + 12x^3 - 36x^2 + 24x \), we first need to factor out the common factors from each term.
Let's analyze each term:
- \( 4x^4 \) has factors of \( 4 \) and \( x^4 \).
- \( 12x^3 \) has factors of \( 12 \) and \( x^3 \).
- \( -36x^2 \) has factors of \( -36 \) and \( x^2 \).
- \( 24x \) has factors of \( 24 \) and \( x \).
Next, we find the GCF of the numerical coefficients: \( 4 \), \( 12 \), \( -36 \), and \( 24 \).
- 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 are \( 1, 2, 4 \). The greatest of these is \( 4 \).
Next, we look at the variable \( x \). The lowest power of \( x \) in the polynomial terms is \( x^1 \) (from the \( 24x \) term). Therefore, the GCF in terms of \( x \) is \( x \).
Now, combining the GCF of the coefficients and the variable part, we get:
GCF = \( 4x \)
Thus, the GCF of the polynomial \( 4x^4 + 12x^3 - 36x^2 + 24x \) is 4x.
The correct response is: 4x.