Question

Question 4(Multiple Choice Worth 2 points)
(02.03 MC)

Which of the following is a factor of x3 − 1331?

x − 11
x2 − 11x + 121
x2 + 22x + 121
None of the above
Question 5(Multiple Choice Worth 2 points)
(02.03 MC)

Which of the following is a factor of 5x3 − 135?

135
x + 3
x2 + 3x + 9
x - 5

Answer 1

To determine the factors of each polynomial, we will use factorization techniques.

**Question 4:**

Given \( x^3 - 1331 \), we can factor it using the difference of cubes formula:

- The formula for difference of cubes is:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, we can let \( a = x \) and \( b = 11 \) since \( 1331 = 11^3 \).

Applying the formula, we get:
\[
x^3 - 1331 = (x - 11)(x^2 + 11x + 121)
\]

Thus, the factors are \( x - 11 \) and \( x^2 + 11x + 121 \).

From the options provided, the correct factor for \( x^3 - 1331 \) is:

**Answer:** \( x - 11 \)

---

**Question 5:**

For \( 5x^3 - 135 \), we can factor out a common factor first.

We notice that both terms share a factor of 5:
\[
5(x^3 - 27)
\]
Next, we can factor \( x^3 - 27 \) using the difference of cubes formula as above:

Let \( a = x \) and \( b = 3 \) (since \( 27 = 3^3 \)):
\[
x^3 - 27 = (x - 3)(x^2 + 3x + 9)
\]

Thus, we have:
\[
5(x^3 - 27) = 5(x - 3)(x^2 + 3x + 9)
\]

Now, the factors of \( 5x^3 - 135 \) include \( x - 3 \) and \( x^2 + 3x + 9 \). Among the options given, the correct factor is:

**Answer:** \( x^2 + 3x + 9 \)