- Recognize \(x^9 - 1000\) as a difference of cubes: \((x^3)^3 - 10^3\).
- Apply the identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\) with \(a = x^3\) and \(b = 10\).
- Simplify to obtain the final expression: \((x^3 - 10)(x^6 + 10x^3 + 100)\).
- Final Answer: \(\boxed{(x^3 - 10)(x^6 + 10x^3 + 100)}\).
Explanation
- Identifying the Structure of the Expression
The expression to factor is \(x^9 - 1000\). To factor this, we observe that both \(x^9\) and \(1000\) are perfect cubes: \(x^9 = (x^3)^3\) and \(1000 = 10^3\). This allows us to recognize the expression as a difference of two cubes, which follows the algebraic identity \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\).
- Factoring the Expression
We set \(a = x^3\) and \(b = 10\). Applying the difference of cubes identity:
\[x^9 - 1000 = (x^3)^3 - 10^3 = (x^3 - 10)((x^3)^2 + (x^3)(10) + 10^2)\]
Now, we simplify the terms within the second factor:
- \((x^3)^2 = x^6\)
- \((x^3)(10) = 10x^3\)
- \(10^2 = 100\)
Substituting these values back, we get the factored form: \((x^3 - 10)(x^6 + 10x^3 + 100)\).
- Conclusion and Selection
Comparing the derived result \((x^3 - 10)(x^6 + 10x^3 + 100)\) with the given options, we see that it matches the first option. The other options are incorrect based on the expansion of the cubes. Therefore, the correct factored form is \((x^3 - 10)(x^6 + 10x^3 + 100)\).
Examples
In fields like computer science or data engineering, identifying patterns in complex expressions is essential for optimization. Just as we decomposed \(x^9 - 1000\) to simplify it, engineers often break down complex functions or data structures into simpler components to perform calculations faster or to find roots of equations. Recognizing algebraic identities like the difference of cubes allows for efficient symbolic manipulation in software development and mathematical modeling.