x3 - 64 using sum and difference of two cubes

To factor the expression \( x^3 - 64 \) using the sum and difference of cubes, we follow these steps:

### Step 1: Identify the cubes
The given expression is \( x^3 - 64 \).

We recognize that:
- \( x^3 \) is the cube of \( x \) (i.e., \( x^3 = (x)^3 \)).
- \( 64 \) is the cube of \( 4 \) (i.e., \( 64 = 4^3 \)).

### Step 2: Apply the difference of cubes formula
The difference of cubes formula is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

In this case, \( a = x \) and \( b = 4 \). Substituting these into the formula:

\[
x^3 - 4^3 = (x - 4)(x^2 + 4x + 16)
\]

### Final Answer
So, the factorization of \( x^3 - 64 \) is:

\[
x^3 - 64 = (x - 4)(x^2 + 4x + 16)
\]

Would you like more details or have any questions?

Here are some related questions you might find helpful:

1. How do you factor the sum of cubes \( x^3 + 27 \)?
2. What is the general formula for factoring a sum or difference of cubes?
3. Can you explain how to recognize when an expression is a sum or difference of cubes?
4. How can you verify the factorization of \( x^3 - 64 \)?
5. What is the next step after factoring a difference of cubes?
6. Are there any real-world applications for factoring cubes in mathematics?
7. How does factoring cubes relate to solving polynomial equations?
8. Can the expression \( x^3 - 64 \) be further factored?

**Tip:** Always check if the expression is a perfect cube before applying the sum or difference of cubes formula.

Learn how to factor the expression x^3 - 64 using the sum and difference of cubes formula with a step-by-step solution. This problem covers algebraic concepts such as cubic equations and factoring techniques. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation