PART 5 - Mathematical Knowledge $$ \begin{array}{l} \text { Factor: } a^{3}+b^{3} \ \text { A }(a+b)\left(a^{2}-a b+b^{2}\right) \ B(a+b)\left(a^{2}-a b-b^{2}\right) \ C(a-b)\left(a^{2}+a b-b^{2}\right) \ D(a+b)(a+b)(a+b)\end{array} $$

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UpStudy AI · Accepted Answer

To factor the expression $$ a^{3} + b^{3} $$, let's analyze each of the given options:

1. **Option A**: $$ (a + b)\left(a^{2} - ab + b^{2}\right) $$
2. **Option B**: $$ (a + b)\left(a^{2} - ab - b^{2}\right) $$
3. **Option C**: $$ (a - b)\left(a^{2} + ab - b^{2}\right) $$
4. **Option D**: $$ (a + b)(a + b)(a + b) $$

### Factoring $$ a^{3} + b^{3} $$:

The sum of cubes formula is a standard algebraic identity:

$$
a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})
$$

Comparing this with the provided options:

- **Option A** exactly matches the sum of cubes formula.

### Verification of Other Options:

- **Option B**: The second factor $$ a^{2} - ab - b^{2} $$ is incorrect. It should be $$ a^{2} - ab + b^{2} $$.
  
- **Option C**: This represents the difference of cubes $$ a^{3} - b^{3} $$, not the sum.
  
- **Option D**: This is the expansion of $$ (a + b)^3 $$, which does not correctly factor $$ a^{3} + b^{3} $$.

### **Conclusion**

The correct factorization of $$ a^{3} + b^{3} $$ is:

$$ \boxed{\text{A }(a+b)\left(a^{2}-a b+b^{2}\right)} $$
The correct factorization of $$ a^{3} + b^{3} $$ is $$ (a + b)(a^{2} - ab + b^{2}) $$.

PART 5 - Mathematical Knowledge

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