At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.

Which of the following is a factor of [tex]$x^3 - 1331$[/tex]?

A. [tex]$x - 11$[/tex]
B. [tex][tex]$x + 11$[/tex][/tex]
C. [tex]$x - 13$[/tex]
D. [tex]$x + 13$[/tex]


Sagot :

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.