To determine the factors of the polynomial [tex]\( x^3 - 1331 \)[/tex], we will follow these logical steps:
1. Step 1: Recognize the polynomial structure.
The given polynomial is [tex]\( x^3 - 1331 \)[/tex]. Notice that 1331 can be written as [tex]\(11^3\)[/tex]. Therefore, the polynomial can be expressed as:
[tex]\[
x^3 - 11^3
\][/tex]
2. Step 2: Identify the difference of cubes formula.
Next, we utilize the difference of cubes formula, which states:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In our problem, [tex]\( a = x \)[/tex] and [tex]\( b = 11 \)[/tex].
3. Step 3: Apply the difference of cubes formula.
Substituting [tex]\( x \)[/tex] for [tex]\( a \)[/tex] and 11 for [tex]\( b \)[/tex] in the difference of cubes formula, we get:
[tex]\[
x^3 - 11^3 = (x - 11)(x^2 + 11x + 121)
\][/tex]
4. Step 4: Identify the factors.
Therefore, the factors of the polynomial [tex]\( x^3 - 1331 \)[/tex] are:
[tex]\[
x - 11 \quad \text{and} \quad x^2 + 11x + 121
\][/tex]
5. Step 5: Answer the multiple-choice question.
Given that the problem is asking for a factor, we conclude that both [tex]\( x - 11 \)[/tex] and [tex]\( x^2 + 11x + 121 \)[/tex] are factors of the polynomial [tex]\( x^3 - 1331 \)[/tex].
Hence, the answer to the question "Which of the following is a factor of [tex]\( x^3 - 1331 \)[/tex]?" is:
[tex]\[
\boxed{x - 11 \quad \text{or} \quad x^2 + 11x + 121}
\][/tex]
Depending on the particular choices presented in the multiple-choice question, you would select the one that matches one of these factors.
