- Identify the expression \(x^3 + 64\) as a sum of cubes \(a^3 + b^3\) where \(a=x\) and \(b=4\).
- Apply the sum of cubes identity \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\).
- Substitute \(a=x\) and \(b=4\) to get the factored form \((x + 4)(x^2 - 4x + 16)\).
- Match the result with the given choices, confirming that option (a) is correct.
- Final Answer: \(\boxed{a}\)
Explanation
- Problem Analysis
The expression we are asked to factor is \(x^3 + 64\). This is a classic algebraic expression that takes the form of a sum of two cubes, represented by the structure \(a^3 + b^3\). Recognizing this form allows us to apply a well-known algebraic identity to factor it efficiently. Our objective is to identify the base components \(a\) and \(b\) and then use the formula to find the correct factored form from the given choices.
- Factoring the Expression
The sum of cubes identity is given by:
\[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
In our expression \(x^3 + 64\), we observe that:
- \(a^3 = x^3\), which implies \(a = x\).
- \(b^3 = 64\). Since \(4 \times 4 \times 4 = 64\), we have \(b = 4\).
Now, substituting \(a = x\) and \(b = 4\) into the identity, we get:
\[x^3 + 4^3 = (x + 4)(x^2 - x(4) + 4^2)\]
Calculating the terms inside the second factor, we find \(x^2 - 4x + 16\). Thus, the fully factored form is:
\[(x + 4)(x^2 - 4x + 16)\]
- Verification and Conclusion
We compare our derived result \((x + 4)(x^2 - 4x + 16)\) with the provided options:
A\((x + 4)(x^2 - 4x + 16)\)
B\((x - 4)(x^2 + 4x + 16)\)
C\((x^2 + 8)(x + 8)\)
The expression matches option (a) exactly. Option (b) corresponds to the difference of cubes \(x^3 - 64\), and option (c) is simply an incorrect distribution. Therefore, the correct factored form is \((x + 4)(x^2 - 4x + 16)\). The final answer is \(\boxed{a}\).
Examples
In fields like computer science and data storage, factoring polynomials like \(x^3 + 64\) can be useful for simplifying complex algorithmic expressions or mapping data indices in 3D spaces. Understanding algebraic identities helps in optimizing calculations, such as reducing the number of operations required to compute an expression by pre-factoring it, which ultimately improves the performance of software programs and mathematical modeling tasks.