$$ \left. \begin{array} { l } { m ^ { 3 } + 27 } \ { ( m + 3 ) ^ { 3 } } \ { ( m + 3 ) ( m ^ { 2 } + 9 ) } \ { ( m - 3 ) ( m ^ { 2 } + 3 m + 9 ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { m ^ { 3 } + 27 } \ { ( m + 3 ) ^ { 3 } } \ { ( m + 3 ) ( m ^ { 2 } + 9 ) } \ { ( m - 3 ) ( m ^ { 2 } + 3 m + 9 ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

It looks like you've provided a set of algebraic expressions involving the variable $$ m $$. Let's analyze each expression to understand their relationships, especially focusing on polynomial factorization.

### 1. $$ m^3 + 27 $$

**Interpretation:** This is a **sum of cubes**, since $$ 27 = 3^3 $$.

**Factorization:**
The sum of cubes formula is:
$$
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
$$
Applying this to $$ m^3 + 27 $$:
$$
m^3 + 27 = m^3 + 3^3 = (m + 3)(m^2 - 3m + 9)
$$

### 2. $$ (m + 3)^3 $$

**Interpretation:** This is the **expansion of a binomial raised to the third power**.

**Expansion:**
Using the binomial theorem or straightforward multiplication:
$$
(m + 3)^3 = m^3 + 3 \cdot m^2 \cdot 3 + 3 \cdot m \cdot 3^2 + 3^3 = m^3 + 9m^2 + 27m + 27
$$
**Comparison:** This is **not equal** to $$ m^3 + 27 $$ unless $$ m = 0 $$.

### 3. $$ (m + 3)(m^2 + 9) $$

**Interpretation:** This appears to be an attempted factorization, but it's **incomplete** for $$ m^3 + 27 $$.

**Multiplication:**
$$
(m + 3)(m^2 + 9) = m^3 + 9m + 3m^2 + 27 = m^3 + 3m^2 + 9m + 27
$$
**Comparison:** This result does **not** match $$ m^3 + 27 $$ unless $$ m = 0 $$. Therefore, this is **not a correct factorization** of $$ m^3 + 27 $$.

### 4. $$ (m - 3)(m^2 + 3m + 9) $$

**Interpretation:** This is the factorization of a **difference of cubes**, specifically $$ m^3 - 27 $$.

**Factorization:**
The difference of cubes formula is:
$$
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
$$
Applying this to $$ m^3 - 27 $$:
$$
m^3 - 27 = m^3 - 3^3 = (m - 3)(m^2 + 3m + 9)
$$

### **Summary of Correct Factorizations:**

1. **Sum of Cubes:**
   $$
   m^3 + 27 = (m + 3)(m^2 - 3m + 9)
   $$
   
2. **Difference of Cubes:**
   $$
   m^3 - 27 = (m - 3)(m^2 + 3m + 9)
   $$
   
3. **Expansion:**
   $$
   (m + 3)^3 = m^3 + 9m^2 + 27m + 27
   $$

4. **Incorrect Factorization Attempt:**
   $$
   (m + 3)(m^2 + 9) \neq m^3 + 27
   $$
   
### **Key Takeaways:**

- **Sum of Cubes:** $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$
- **Difference of Cubes:** $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$
- **Cubing a Binomial:** Expanding $$ (a + b)^3 $$ results in $$ a^3 + 3a^2b + 3ab^2 + b^3 $$, which is generally different from just $$ a^3 + b^3 $$.

Ensure you use the correct formulas when factoring or expanding polynomials to avoid inconsistencies.
$$ m^3 + 27 = (m + 3)(m^2 - 3m + 9) $$ and $$ m^3 - 27 = (m - 3)(m^2 + 3m + 9) $$.

\( \left. \begin{array} { l } { m ^ { 3 } + 27 } \\ { ( m + 3 ) ^ { 3 } } \\ { ( m + 3 ) ( m ^ { 2 } + 9 ) } \\ { ( m - 3 ) ( m ^ { 2 } + 3 m + 9 ) } \end{array} \right. \)

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