Question

Question 1(Multiple Choice Worth 2 points)
(02.03 MC)

Which of the following is a factor of 27x3 + 512y3?

3
3x − 8y
9x2 − 24xy + 64y2
9x2 + 24xy + 64y2
Question 2(Multiple Choice Worth 2 points)
(02.03 MC)

Which of the following is a factor of x15 − 64?

x5 − 4
x10 − 4x5 + 16
x10 − 4
x15 + 8x + 16

Answer 1

**Question 1:**

To factor \( 27x^3 + 512y^3 \), we can recognize this expression as a sum of cubes. The formula for the sum of cubes \( a^3 + b^3 \) can be factored as:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

In our case:
- \( a = 3x \) (since \( (3x)^3 = 27x^3 \))
- \( b = 8y \) (since \( (8y)^3 = 512y^3 \))

Using the sum of cubes formula, we have:
\[
27x^3 + 512y^3 = (3x + 8y)((3x)^2 - (3x)(8y) + (8y)^2)
\]
Calculating the second part:
\[
= (3x + 8y)(9x^2 - 24xy + 64y^2)
\]

From the options, one of the factors presented is \( 3x - 8y \) which does not match, and the cubic term \( 9x^2 - 24xy + 64y^2 \).

Thus, **the correct answer** is:
**9x² - 24xy + 64y²**

---

**Question 2:**

To factor \( x^{15} - 64 \), we recognize this as a difference of cubes. The formula for the difference of cubes \( a^3 - b^3 \) is given as:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, we can express \( x^{15} \) as \( (x^5)^3 \) and \( 64 \) as \( 4^3 \):
- \( a = x^5 \)
- \( b = 4 \)

Applying the difference of cubes formula:
\[
x^{15} - 64 = (x^5 - 4)((x^5)^2 + (x^5)(4) + 4^2)
\]
Simplifying the second factor:
\[
= (x^5 - 4)(x^{10} + 4x^5 + 16)
\]

From the options, the first factor is \( x^5 - 4 \), which indeed is a factor.

Thus, **the correct answer** is:
**x5 - 4**