To determine if \( (x + 2) \) is a factor of the polynomial \( f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36 \), we can use the Remainder Theorem. According to the theorem, if \( (x - c) \) is a factor of \( f(x) \), then \( f(c) = 0 \).
Since we have \( (x + 2) \), we can set \( c = -2 \). We will evaluate \( f(-2) \).
The polynomial we have is:
\[ f(x) = x^5 - 3x^4 - 9x^3 + 23x^2 + 24x - 36 \]
Now we need to calculate \( f(-2) \):
\[ f(-2) = (-2)^5 - 3(-2)^4 - 9(-2)^3 + 23(-2)^2 + 24(-2) - 36 \]
Calculating each term:
- \( (-2)^5 = -32 \)
- \( -3(-2)^4 = -3(16) = -48 \)
- \( -9(-2)^3 = -9(-8) = 72 \)
- \( 23(-2)^2 = 23(4) = 92 \)
- \( 24(-2) = -48 \)
- The constant term is \( -36 \)
Now substituting these values into the equation:
\[ f(-2) = -32 - 48 + 72 + 92 - 48 - 36 \]
Now, perform the addition and subtraction step by step:
\[ f(-2) = -32 - 48 = -80 \]
\[ f(-2) = -80 + 72 = -8 \]
\[ f(-2) = -8 + 92 = 84 \]
\[ f(-2) = 84 - 48 = 36 \]
\[ f(-2) = 36 - 36 = 0 \]
Since \( f(-2) = 0 \), this means that \( (x + 2) \) is indeed a factor of the polynomial \( f(x) \).
Thus, the conclusion is:
\[ \text{Yes, } (x + 2) \text{ is a factor of } f(x). \]