Question

which of the following is a factor of (6x^3 + 6)?

• (x - 1)
• (x + 1)
• (x^2 - 2x + 1)
• (x^2 + x + 1)

question 4 (multiple choice worth 2 points)
(02.03r mc)

which of the following is a factor of (24x^6 - 1029y^3)?

• (24)
• (2x^2 + 7y)
• (4x^4 + 14x^2y + 49y^2)
• all of the above

Explanation:

Factor out the greatest common factor of the first expression

We begin by factoring the first polynomial expression:

$$6x^3 + 6$$

Using the Greatest Common Factor knowledge point, we factor out \(6\):

$$6(x^3 + 1)$$

Apply the sum of cubes formula to the first expression

Next, we factor the sum of cubes term \(x^3 + 1\).
Using the Sum and Difference of Cubes knowledge point, we apply the identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\) where \(a = x\) and \(b = 1\):

$$x^3 + 1 = (x + 1)(x^2 - x + 1)$$

Thus, the completely factored form of the first expression is:

$$6(x + 1)(x^2 - x + 1)$$

Comparing this with the given options, \(x + 1\) is a factor.

Factor out the greatest common factor of the second expression

Now we factor the second polynomial expression:

$$24x^6 - 1029y^3$$

Using the Greatest Common Factor knowledge point, we find the greatest common factor of \(24\) and \(1029\), which is \(3\):

$$3(8x^6 - 343y^3)$$

Apply the difference of cubes formula to the second expression

We rewrite the terms inside the parentheses as perfect cubes:

$$8x^6 = (2x^2)^3$$

$$343y^3 = (7y)^3$$

Using the Sum and Difference of Cubes knowledge point, we apply the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) where \(a = 2x^2\) and \(b = 7y\):

$$8x^6 - 343y^3 = (2x^2 - 7y)((2x^2)^2 + (2x^2)(7y) + (7y)^2)$$

$$8x^6 - 343y^3 = (2x^2 - 7y)(4x^4 + 14x^2y + 49y^2)$$

Thus, the completely factored form of the second expression is:

$$3(2x^2 - 7y)(4x^4 + 14x^2y + 49y^2)$$

Identify the correct option for the second expression

Using the Factoring Polynomials Completely knowledge point, we analyze the factors of the second expression:

• \(24\) is not a factor because only \(3\) was factored out.
• \(2x^2 + 7y\) is not a factor (the binomial factor is \(2x^2 - 7y\)).
• \(4x^4 + 14x^2y + 49y^2\) is a factor.

Therefore, the correct choice is the third option.

Answer:

Question 1

• (A) \(x - 1\)
• (B) \(x + 1\) (Correct answer)
• (C) \(x^2 - 2x + 1\)
• (D) \(x^2 + x + 1\)

Question 2

• (A) 24
• (B) \(2x^2 + 7y\)
• (C) \(4x^4 + 14x^2y + 49y^2\) (Correct answer)
• (D) All of the above