Factor the given using Difference of Two cubes: m⁶ - 1 000

To factor the given expression \( m^6 - 1000 \) using the difference of two cubes, follow these steps:

### Step 1: Identify the cubes
The expression can be written as a difference of two cubes:

\[
m^6 - 1000 = (m^2)^3 - 10^3
\]

Here:
- \( a^3 = (m^2)^3 \)
- \( b^3 = 10^3 \)

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

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

Using this formula, we substitute \( a = m^2 \) and \( b = 10 \):

\[
m^6 - 1000 = (m^2 - 10)\left((m^2)^2 + (m^2)(10) + 10^2\right)
\]

### Step 3: Simplify the expression
Now, simplify the expression inside the second set of parentheses:

\[
m^6 - 1000 = (m^2 - 10)\left(m^4 + 10m^2 + 100\right)
\]

So, the factored form of \( m^6 - 1000 \) is:

\[
\boxed{(m^2 - 10)(m^4 + 10m^2 + 100)}
\]

Would you like to explore this further or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How do you factor \( x^6 - 27 \) using the difference of cubes?
2. Can the expression \( m^6 + 1000 \) be factored similarly?
3. What are the possible roots of \( m^6 - 1000 \) in the factored form?
4. How would you factor \( m^6 - 64 \) using the difference of squares and cubes?
5. What is the importance of recognizing patterns like the difference of cubes in algebra?

**Tip:** Always check if a given polynomial can be factored using common identities like difference of squares or cubes, as it simplifies solving equations significantly.

Learn how to factor the polynomial m^6 - 1000 using the difference of cubes formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including difference of cubes, polynomial factoring, and algebraic simplification. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation