Factoring Sums and Differences of Cubes Skill Sheet with answer key and QR code.

The image you provided is an answer key for factoring sums and differences of cubes. The problems involve factoring expressions that are either the sum or difference of two cubes. Below, I will explain the general method for factoring sums and differences of cubes and then verify the solutions given in the answer key.

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General Formulas for Factoring Sums and Differences of Cubes

1. Difference of Cubes:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

2. Sum of Cubes:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

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Step-by-Step Verification of Each Problem

#### Problem 1: \( x^3 - 8 \)
- This is a difference of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = x^3 \quad \text{and} \quad b^3 = 8 \implies a = x \quad \text{and} \quad b = 2
\]
- Apply the formula for the difference of cubes:
\[
x^3 - 8 = (x - 2)(x^2 + 2x + 4)
\]
- Answer: \((x - 2)(x^2 + 2x + 4)\)

#### Problem 2: \( x^3 + 8 \)
- This is a sum of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = x^3 \quad \text{and} \quad b^3 = 8 \implies a = x \quad \text{and} \quad b = 2
\]
- Apply the formula for the sum of cubes:
\[
x^3 + 8 = (x + 2)(x^2 - 2x + 4)
\]
- Answer: \((x + 2)(x^2 - 2x + 4)\)

#### Problem 3: \( x^3 - 64 \)
- This is a difference of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = x^3 \quad \text{and} \quad b^3 = 64 \implies a = x \quad \text{and} \quad b = 4
\]
- Apply the formula for the difference of cubes:
\[
x^3 - 64 = (x - 4)(x^2 + 4x + 16)
\]
- Answer: \((x - 4)(x^2 + 4x + 16)\)

#### Problem 4: \( 27x^3 + 125 \)
- This is a sum of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = 27x^3 \quad \text{and} \quad b^3 = 125 \implies a = 3x \quad \text{and} \quad b = 5
\]
- Apply the formula for the sum of cubes:
\[
27x^3 + 125 = (3x + 5)((3x)^2 - (3x)(5) + 5^2)
\]
Simplify:
\[
27x^3 + 125 = (3x + 5)(9x^2 - 15x + 25)
\]
- Answer: \((3x + 5)(9x^2 - 15x + 25)\)

#### Problem 5: \( y^3 - 64 \)
- This is a difference of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = y^3 \quad \text{and} \quad b^3 = 64 \implies a = y \quad \text{and} \quad b = 4
\]
- Apply the formula for the difference of cubes:
\[
y^3 - 64 = (y - 4)(y^2 + 4y + 16)
\]
- Answer: \((y - 4)(y^2 + 4y + 16)\)

#### Problem 6: \( x^3y^3 + 27 \)
- This is a sum of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = x^3y^3 \quad \text{and} \quad b^3 = 27 \implies a = xy \quad \text{and} \quad b = 3
\]
- Apply the formula for the sum of cubes:
\[
x^3y^3 + 27 = (xy + 3)((xy)^2 - (xy)(3) + 3^2)
\]
Simplify:
\[
x^3y^3 + 27 = (xy + 3)(x^2y^2 - 3xy + 9)
\]
- Answer: \((xy + 3)(x^2y^2 - 3xy + 9)\)

#### Problem 7: \( 8x^3 - 216 \)
- This is a difference of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = 8x^3 \quad \text{and} \quad b^3 = 216 \implies a = 2x \quad \text{and} \quad b = 6
\]
- Apply the formula for the difference of cubes:
\[
8x^3 - 216 = (2x - 6)((2x)^2 + (2x)(6) + 6^2)
\]
Simplify:
\[
8x^3 - 216 = (2x - 6)(4x^2 + 12x + 36)
\]
- Answer: \((2x - 6)(4x^2 + 12x + 36)\)

#### Problem 8: \( x^3 - y^3 \)
- This is a difference of cubes.
- Identify \( a \) and \( b \):
\[
a^3 = x^3 \quad \text{and} \quad b^3 = y^3 \implies a = x \quad \text{and} \quad b = y
\]
- Apply the formula for the difference of cubes:
\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]
- Answer: \((x - y)(x^2 + xy + y^2)\)

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Final Answer

The solutions provided in the answer key are correct. Here they are summarized:

1. \( x^3 - 8 = (x - 2)(x^2 + 2x + 4) \)
2. \( x^3 + 8 = (x + 2)(x^2 - 2x + 4) \)
3. \( x^3 - 64 = (x - 4)(x^2 + 4x + 16) \)
4. \( 27x^3 + 125 = (3x + 5)(9x^2 - 15x + 25) \)
5. \( y^3 - 64 = (y - 4)(y^2 + 4y + 16) \)
6. \( x^3y^3 + 27 = (xy + 3)(x^2y^2 - 3xy + 9) \)
7. \( 8x^3 - 216 = (2x - 6)(4x^2 + 12x + 36) \)
8. \( x^3 - y^3 = (x - y)(x^2 + xy + y^2) \)

\boxed{\text{All answers are correct.}}