Mathematics worksheet for Grade 8 students on factoring sum or difference of two cubes, provided by Heisenberg Tutorial Center.

Problem: Factor each expression completely.

The task involves factoring expressions that are sums or differences of two cubes. The formulas for factoring sums and differences of two cubes are as follows:

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

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

We will use these formulas to factor each given expression.

---

Step-by-Step Solutions

#### 1. \( x^3 + 8 \)
This is a sum of two cubes:
\[
x^3 + 8 = x^3 + 2^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = x \) and \( b = 2 \):
\[
x^3 + 8 = (x + 2)(x^2 - x \cdot 2 + 2^2) = (x + 2)(x^2 - 2x + 4)
\]

Answer:
\[
\boxed{(x + 2)(x^2 - 2x + 4)}
\]

---

#### 2. \( a^3 + 64 \)
This is a sum of two cubes:
\[
a^3 + 64 = a^3 + 4^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = a \) and \( b = 4 \):
\[
a^3 + 64 = (a + 4)(a^2 - a \cdot 4 + 4^2) = (a + 4)(a^2 - 4a + 16)
\]

Answer:
\[
\boxed{(a + 4)(a^2 - 4a + 16)}
\]

---

#### 3. \( a^3 + 216 \)
This is a sum of two cubes:
\[
a^3 + 216 = a^3 + 6^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = a \) and \( b = 6 \):
\[
a^3 + 216 = (a + 6)(a^2 - a \cdot 6 + 6^2) = (a + 6)(a^2 - 6a + 36)
\]

Answer:
\[
\boxed{(a + 6)(a^2 - 6a + 36)}
\]

---

#### 4. \( 27 + 8x^3 \)
This is a sum of two cubes:
\[
27 + 8x^3 = 3^3 + (2x)^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = 3 \) and \( b = 2x \):
\[
27 + 8x^3 = (3 + 2x)(3^2 - 3 \cdot 2x + (2x)^2) = (3 + 2x)(9 - 6x + 4x^2)
\]

Answer:
\[
\boxed{(3 + 2x)(9 - 6x + 4x^2)}
\]

---

#### 5. \( a^3 - 216 \)
This is a difference of two cubes:
\[
a^3 - 216 = a^3 - 6^3
\]
Using the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = a \) and \( b = 6 \):
\[
a^3 - 216 = (a - 6)(a^2 + a \cdot 6 + 6^2) = (a - 6)(a^2 + 6a + 36)
\]

Answer:
\[
\boxed{(a - 6)(a^2 + 6a + 36)}
\]

---

#### 6. \( 64x^3 - 27 \)
This is a difference of two cubes:
\[
64x^3 - 27 = (4x)^3 - 3^3
\]
Using the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 4x \) and \( b = 3 \):
\[
64x^3 - 27 = (4x - 3)((4x)^2 + (4x)(3) + 3^2) = (4x - 3)(16x^2 + 12x + 9)
\]

Answer:
\[
\boxed{(4x - 3)(16x^2 + 12x + 9)}
\]

---

#### 7. \( 27m^3 - 125 \)
This is a difference of two cubes:
\[
27m^3 - 125 = (3m)^3 - 5^3
\]
Using the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = 3m \) and \( b = 5 \):
\[
27m^3 - 125 = (3m - 5)((3m)^2 + (3m)(5) + 5^2) = (3m - 5)(9m^2 + 15m + 25)
\]

Answer:
\[
\boxed{(3m - 5)(9m^2 + 15m + 25)}
\]

---

#### 8. \( x^3 - 64 \)
This is a difference of two cubes:
\[
x^3 - 64 = x^3 - 4^3
\]
Using the difference of cubes formula:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( a = x \) and \( b = 4 \):
\[
x^3 - 64 = (x - 4)(x^2 + x \cdot 4 + 4^2) = (x - 4)(x^2 + 4x + 16)
\]

Answer:
\[
\boxed{(x - 4)(x^2 + 4x + 16)}
\]

---

#### 9. \( 432 + 250m^3 \)
First, factor out the greatest common factor (GCF):
\[
432 + 250m^3 = 2(216 + 125m^3)
\]
Now, factor \( 216 + 125m^3 \) as a sum of two cubes:
\[
216 + 125m^3 = 6^3 + (5m)^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = 6 \) and \( b = 5m \):
\[
216 + 125m^3 = (6 + 5m)(6^2 - 6 \cdot 5m + (5m)^2) = (6 + 5m)(36 - 30m + 25m^2)
\]
Thus:
\[
432 + 250m^3 = 2(6 + 5m)(36 - 30m + 25m^2)
\]

Answer:
\[
\boxed{2(6 + 5m)(36 - 30m + 25m^2)}
\]

---

#### 10. \( 81x^3 + 192 \)
First, factor out the greatest common factor (GCF):
\[
81x^3 + 192 = 3(27x^3 + 64)
\]
Now, factor \( 27x^3 + 64 \) as a sum of two cubes:
\[
27x^3 + 64 = (3x)^3 + 4^3
\]
Using the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
Here, \( a = 3x \) and \( b = 4 \):
\[
27x^3 + 64 = (3x + 4)((3x)^2 - (3x)(4) + 4^2) = (3x + 4)(9x^2 - 12x + 16)
\]
Thus:
\[
81x^3 + 192 = 3(3x + 4)(9x^2 - 12x + 16)
\]

Answer:
\[
\boxed{3(3x + 4)(9x^2 - 12x + 16)}
\]

---

Final Answers

\[
\boxed{
\begin{aligned}
1. & \ (x + 2)(x^2 - 2x + 4) \\
2. & \ (a + 4)(a^2 - 4a + 16) \\
3. & \ (a + 6)(a^2 - 6a + 36) \\
4. & \ (3 + 2x)(9 - 6x + 4x^2) \\
5. & \ (a - 6)(a^2 + 6a + 36) \\
6. & \ (4x - 3)(16x^2 + 12x + 9) \\
7. & \ (3m - 5)(9m^2 + 15m + 25) \\
8. & \ (x - 4)(x^2 + 4x + 16) \\
9. & \ 2(6 + 5m)(36 - 30m + 25m^2) \\
10. & \ 3(3x + 4)(9x^2 - 12x + 16)
\end{aligned}
}
\]