Question

which of the following is a factor of (1331x^3 - 8y^3)?

• 4
• (11x - 2y)
• (121x^2 + 44xy + 4y^2)
• (121x^2 - 22xy + 4y^2)

question 2 (multiple choice worth 2 points)
(02.03r mc)

which of the following is a factor of (x^6 + 1000)?

• (x^2)
• (x^2 - 10)
• (x^4 - 10x^2 + 100)
• (x^4 + 10x^2 + 100)

Explanation:

Identify the algebraic structures

We analyze the expressions to recognize special factoring patterns.
The first expression is \(1331x^3 - 8y^3\), which is a difference of cubes.
The second expression is \(x^6 + 1000\), which is a sum of cubes.

Apply difference of cubes formula

Using the Factoring Polynomials Completely knowledge point:
The difference of cubes formula is:

$$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$

For \(1331x^3 - 8y^3\), we identify:

$$a = 11x,\quad b = 2y$$

Substituting these values into the formula:

$$1331x^3 - 8y^3 = (11x - 2y)((11x)^2 + (11x)(2y) + (2y)^2)$$

$$1331x^3 - 8y^3 = (11x - 2y)(121x^2 + 22xy + 4y^2)$$

Thus, \(11x - 2y\) is a factor.

Apply sum of cubes formula

Using the Factoring Polynomials Completely knowledge point:
The sum of cubes formula is:

$$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$

For \(x^6 + 1000\), we rewrite the terms as cubes:

$$a = x^2,\quad b = 10$$

Substituting these values into the formula:

$$x^6 + 1000 = (x^2 + 10)((x^2)^2 - (x^2)(10) + 10^2)$$

$$x^6 + 1000 = (x^2 + 10)(x^4 - 10x^2 + 100)$$

Thus, \(x^4 - 10x^2 + 100\) is a factor.

Answer:

Question 1

• (A) 4
• (B) \(11x - 2y\) (Correct answer)
• (C) \(121x^2 + 44xy + 4y^2\)
• (D) \(121x^2 - 22xy + 4y^2\)

Question 2

• (A) \(x^2\)
• (B) \(x^2 - 10\)
• (C) \(x^4 - 10x^2 + 100\) (Correct answer)
• (D) \(x^4 + 10x^2 + 100\)