To determine which of the given options is a factor of \( x^3 - 1331 \), we can start by rewriting \( 1331 \) as \( 11^3 \). Therefore, we have:
\[ x^3 - 1331 = x^3 - 11^3 \]
This expression can be factored using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
In our case, \( a = x \) and \( b = 11 \), so we can apply the formula:
\[ x^3 - 11^3 = (x - 11)(x^2 + 11x + 121) \]
Now, we will analyze the provided options:
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\( x - 11 \): This is one of the factors we found, \( (x - 11) \).
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\( x^2 - 11x + 121 \): We need to check if this is a factor. Notice that it does not appear in our factorization.
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\( x^2 + 22x + 121 \): Again, this does not appear in our factorization as well.
Since our factorization shows that \( x - 11 \) is indeed a factor of \( x^3 - 1331 \), we can conclude:
The answer is \( x - 11 \).