- Recognize \(b^3 - 1000\) as a difference of cubes: \(b^3 - 10^3\).
- Apply the identity \(a^3 - n^3 = (a - n)(a^2 + an + n^2)\) with \(a=b\) and \(n=10\).
- Substitute the values into the formula to get \((b - 10)(b^2 + 10b + 100)\).
- Conclude that the correct option is \(\boxed{(b - 10)(b^2 + 10b + 100)}\).
Explanation
- Analyzing the Expression
We are tasked with finding the factored form of the expression \(b^3 - 1000\). This expression is a classic example of a "difference of cubes," which follows the algebraic identity \(a^3 - n^3 = (a - n)(a^2 + an + n^2)\). By identifying the components of our expression with the standard identity, we can determine the correct factors.
- Identifying Components for the Identity
In the given expression \(b^3 - 1000\), we observe that \(b^3\) is a cube and \(1000\) is also a perfect cube, since \(10^3 = 1000\). Thus, we can rewrite the expression as:
\[b^3 - 10^3\]
Matching this with the identity \(a^3 - n^3 = (a - n)(a^2 + an + n^2)\), we set \(a = b\) and \(n = 10\).
- Factoring the Expression
Now we substitute \(a = b\) and \(n = 10\) into the identity:
- The first factor \((a - n)\) becomes \((b - 10)\).
- The second factor \((a^2 + an + n^2)\) becomes \((b^2 + b \times 10 + 10^2)\).
Simplifying the terms in the second factor, we get \(b^2 + 10b + 100\).
Therefore, the factored form is \((b - 10)(b^2 + 10b + 100)\).
- Selecting the Correct Option
Comparing our result \((b - 10)(b^2 + 10b + 100)\) with the given options, it is clear that the second option is the correct one.
- Option 1: \((b + 10)(b^2 - 10b + 100)\)
- Option 2: \((b - 10)(b^2 + 10b + 100)\)
- Option 3: \((b - 10)(10b^2 + b + 100)\)
- Option 4: \((b + 10)(10b^2 - b + 100)\)
The factored form is \(\boxed{(b - 10)(b^2 + 10b + 100)}\).
Examples
Understanding how to factor differences of cubes is essential for simplifying complex algebraic expressions, solving higher-degree polynomial equations, and performing division in algebra. For instance, in engineering or computer science, you might encounter such expressions when optimizing algorithms for computational efficiency or simplifying circuit design equations. Mastering this identity allows you to quickly recognize patterns that would otherwise be difficult to compute manually.