Factoring Cubes Challenge

Factor the expression: x^3 + 27

• (x + 9)(x^2 - 9x + 81)

• (x + 3)(x^2 - 3x + 9)

• (x - 3)(x^2 + 3x + 9)

• (x + 1)(x^2 + 1)

Answer explanation

The expression x^3 + 27 is a sum of cubes, which factors as (x + 3)(x^2 - 3x + 9). This matches the correct choice, confirming that (x + 3)(x^2 - 3x + 9) is the valid factorization.

Factor the expression: 8y^3 - 1

• (2y + 1)(4y^2 - 2y + 1)

• (8y - 1)(y^2 + 1)

• (4y - 1)(2y^2 + 2y + 1)

• (2y - 1)(4y^2 + 2y + 1)

Answer explanation

To factor 8y^3 - 1, recognize it as a difference of cubes: (2y)^3 - 1^3. Using the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2), we get (2y - 1)(4y^2 + 2y + 1), which matches the correct choice.

Factor the expression: a^3 + 64

• (a + 4)(a^2 - 4a + 16)

• (a + 4)(a^2 + 4a + 16)

• (a + 8)(a^2 - 8a + 16)

• (a + 2)(a^2 - 2a + 32)

Answer explanation

The expression a^3 + 64 is a sum of cubes, which factors as (a + 4)(a^2 - 4a + 16). This matches the correct choice, as it follows the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2) with a = a and b = 4.

Factor the expression: 125 - b^3

• (5 - b)(25 + 5b + b^2)

• (5 + b)(25 + 5b - b^2)

• (5 - b)(25 - 5b + b^2)

• (5 + b)(25 - 5b + b^2)

Answer explanation

The expression 125 - b^3 is a difference of cubes, which factors as (a - b)(a^2 + ab + b^2) where a = 5 and b = b. Thus, it factors to (5 - b)(25 + 5b + b^2), making this the correct choice.

Factor the expression: 27x^3 + 1

• (3x - 1)(9x^2 + 3x + 1)

• (27x + 1)(x^2 - 1)

• (9x + 3)(3x^2 - 1)

• (3x + 1)(9x^2 - 3x + 1)

Answer explanation

To factor 27x^3 + 1, recognize it as a sum of cubes: (3x)^3 + 1^3. Using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2), we get (3x + 1)(9x^2 - 3x + 1). Thus, the correct choice is (3x + 1)(9x^2 - 3x + 1).

Factor the expression: 64y^3 - 8

• 4(4y^3 - 2)

• 16(4y^3 - 1)

• 8(2y - 1)(4y^2 + 2y + 1)

• 8(2y + 1)(4y^2 - 2y + 1)

Answer explanation

To factor 64y^3 - 8, recognize it as a difference of cubes: (4y)^3 - 2^3. This factors to (4y - 2)(16y^2 + 8y + 4). Further simplifying gives 8(2y - 1)(4y^2 + 2y + 1), which is the correct choice.

Factor the expression: m^3 + 1

• (m + 1)(m^2 - m + 1)

• (m + 2)(m^2 - 2m + 4)

• (m^3 - 1)(m + 1)

• (m - 1)(m^2 + m + 1)

Answer explanation

To factor m^3 + 1, we use the sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). Here, a = m and b = 1, giving us (m + 1)(m^2 - m + 1) as the correct factorization.

Factor the expression: 1 - n^3

• (1 + n)(1 - n + n^2)

• (1 - n)(1 + n + n^2)

• (1 - n^2)(1 + n)

• (1 - n)(1 + n - n^2)

Answer explanation

To factor 1 - n^3, recognize it as a difference of cubes: 1^3 - n^3. This factors to (1 - n)(1^2 + 1*n + n^2) = (1 - n)(1 + n + n^2), making the correct choice (1 - n)(1 + n + n^2).

Factor the expression: 8x^3 + 27

• (8x + 27)(1)

• (2x + 3)(4x^2 - 6x + 9)

• (4x + 3)(2x^2 - 3x + 1)

• (2x - 3)(4x^2 + 6x + 9)

Answer explanation

The expression 8x^3 + 27 is a sum of cubes, which factors as (a + b)(a^2 - ab + b^2) where a = 2x and b = 3. Thus, it factors to (2x + 3)(4x^2 - 6x + 9), making this the correct choice.

Factor the expression: p^3 - 27

• (p - 3)(p^2 - 3p - 9)

• (p + 3)(p^2 - 3p + 9)

• (p - 9)(p^2 + 3p + 3)

• (p - 3)(p^2 + 3p + 9)

Answer explanation

To factor p^3 - 27, recognize it as a difference of cubes: p^3 - 3^3. This factors to (p - 3)(p^2 + 3p + 9), making (p - 3)(p^2 + 3p + 9) the correct choice.