Factoring A Sum+Difference of Cubes - Kuta Software - Free Printable

Problem: Factoring a Sum or Difference of Cubes

The task is to factor each expression completely using the formulas for the sum of cubes and difference of cubes. The formulas are:

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

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

We will apply these formulas to each given expression.

---

Solutions

#### 1. \( x^3 + 125 \)
This is a sum of cubes:
\[
x^3 + 125 = x^3 + 5^3
\]
Using the sum of cubes formula:
\[
a = x, \quad b = 5
\]
\[
x^3 + 5^3 = (x + 5)(x^2 - x \cdot 5 + 5^2) = (x + 5)(x^2 - 5x + 25)
\]
Answer:
\[
\boxed{(x + 5)(x^2 - 5x + 25)}
\]

#### 2. \( a^3 + 64 \)
This is a sum of cubes:
\[
a^3 + 64 = a^3 + 4^3
\]
Using the sum of cubes formula:
\[
a = a, \quad b = 4
\]
\[
a^3 + 4^3 = (a + 4)(a^2 - a \cdot 4 + 4^2) = (a + 4)(a^2 - 4a + 16)
\]
Answer:
\[
\boxed{(a + 4)(a^2 - 4a + 16)}
\]

#### 3. \( x^3 - 64 \)
This is a difference of cubes:
\[
x^3 - 64 = x^3 - 4^3
\]
Using the difference of cubes formula:
\[
a = x, \quad b = 4
\]
\[
x^3 - 4^3 = (x - 4)(x^2 + x \cdot 4 + 4^2) = (x - 4)(x^2 + 4x + 16)
\]
Answer:
\[
\boxed{(x - 4)(x^2 + 4x + 16)}
\]

#### 4. \( u^3 + 8 \)
This is a sum of cubes:
\[
u^3 + 8 = u^3 + 2^3
\]
Using the sum of cubes formula:
\[
a = u, \quad b = 2
\]
\[
u^3 + 2^3 = (u + 2)(u^2 - u \cdot 2 + 2^2) = (u + 2)(u^2 - 2u + 4)
\]
Answer:
\[
\boxed{(u + 2)(u^2 - 2u + 4)}
\]

#### 5. \( x^3 - 27 \)
This is a difference of cubes:
\[
x^3 - 27 = x^3 - 3^3
\]
Using the difference of cubes formula:
\[
a = x, \quad b = 3
\]
\[
x^3 - 3^3 = (x - 3)(x^2 + x \cdot 3 + 3^2) = (x - 3)(x^2 + 3x + 9)
\]
Answer:
\[
\boxed{(x - 3)(x^2 + 3x + 9)}
\]

#### 6. \( 125 - x^3 \)
This is a difference of cubes:
\[
125 - x^3 = 5^3 - x^3
\]
Using the difference of cubes formula:
\[
a = 5, \quad b = x
\]
\[
5^3 - x^3 = (5 - x)(5^2 + 5 \cdot x + x^2) = (5 - x)(25 + 5x + x^2)
\]
Answer:
\[
\boxed{(5 - x)(25 + 5x + x^2)}
\]

#### 7. \( 1 - a^3 \)
This is a difference of cubes:
\[
1 - a^3 = 1^3 - a^3
\]
Using the difference of cubes formula:
\[
a = 1, \quad b = a
\]
\[
1^3 - a^3 = (1 - a)(1^2 + 1 \cdot a + a^2) = (1 - a)(1 + a + a^2)
\]
Answer:
\[
\boxed{(1 - a)(1 + a + a^2)}
\]

#### 8. \( a^3 + 125 \)
This is a sum of cubes:
\[
a^3 + 125 = a^3 + 5^3
\]
Using the sum of cubes formula:
\[
a = a, \quad b = 5
\]
\[
a^3 + 5^3 = (a + 5)(a^2 - a \cdot 5 + 5^2) = (a + 5)(a^2 - 5a + 25)
\]
Answer:
\[
\boxed{(a + 5)(a^2 - 5a + 25)}
\]

#### 9. \( x^3 + 27 \)
This is a sum of cubes:
\[
x^3 + 27 = x^3 + 3^3
\]
Using the sum of cubes formula:
\[
a = x, \quad b = 3
\]
\[
x^3 + 3^3 = (x + 3)(x^2 - x \cdot 3 + 3^2) = (x + 3)(x^2 - 3x + 9)
\]
Answer:
\[
\boxed{(x + 3)(x^2 - 3x + 9)}
\]

#### 10. \( x^3 + 1 \)
This is a sum of cubes:
\[
x^3 + 1 = x^3 + 1^3
\]
Using the sum of cubes formula:
\[
a = x, \quad b = 1
\]
\[
x^3 + 1^3 = (x + 1)(x^2 - x \cdot 1 + 1^2) = (x + 1)(x^2 - x + 1)
\]
Answer:
\[
\boxed{(x + 1)(x^2 - x + 1)}
\]

#### 11. \( 8x^3 + 27 \)
This is a sum of cubes:
\[
8x^3 + 27 = (2x)^3 + 3^3
\]
Using the sum of cubes formula:
\[
a = 2x, \quad b = 3
\]
\[
(2x)^3 + 3^3 = (2x + 3)((2x)^2 - (2x)(3) + 3^2) = (2x + 3)(4x^2 - 6x + 9)
\]
Answer:
\[
\boxed{(2x + 3)(4x^2 - 6x + 9)}
\]

#### 12. \( -27u^3 + 125 \)
This is a difference of cubes:
\[
-27u^3 + 125 = 125 - 27u^3 = 5^3 - (3u)^3
\]
Using the difference of cubes formula:
\[
a = 5, \quad b = 3u
\]
\[
5^3 - (3u)^3 = (5 - 3u)(5^2 + 5 \cdot 3u + (3u)^2) = (5 - 3u)(25 + 15u + 9u^2)
\]
Answer:
\[
\boxed{(5 - 3u)(25 + 15u + 9u^2)}
\]

---

Final Answers

\[
\boxed{
\begin{aligned}
1. & \ (x + 5)(x^2 - 5x + 25) \\
2. & \ (a + 4)(a^2 - 4a + 16) \\
3. & \ (x - 4)(x^2 + 4x + 16) \\
4. & \ (u + 2)(u^2 - 2u + 4) \\
5. & \ (x - 3)(x^2 + 3x + 9) \\
6. & \ (5 - x)(25 + 5x + x^2) \\
7. & \ (1 - a)(1 + a + a^2) \\
8. & \ (a + 5)(a^2 - 5a + 25) \\
9. & \ (x + 3)(x^2 - 3x + 9) \\
10. & \ (x + 1)(x^2 - x + 1) \\
11. & \ (2x + 3)(4x^2 - 6x + 9) \\
12. & \ (5 - 3u)(25 + 15u + 9u^2)
\end{aligned}
}
\]