Verified Answer :

  • Identify the difference of cubes identity: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). For \(8^3 - 2^3\), substitute \(a=8, b=2\) to get \(6 \times 84 = 504\).
  • Identify the sum of cubes identity: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). For \(10^3 + 5^3\), substitute \(a=10, b=5\) to get \(15 \times 75 = 1125\).
  • Final answers are \(\boxed{504}\) and \(\boxed{1125}\).

Explanation

  1. Analysis of the Problem and Identities

The objective is to simplify two numerical expressions, \(8^3 - 2^3\) and \(10^3 + 5^3\), using polynomial identities for the sum and difference of cubes. Polynomial identities are algebraic formulas that allow us to expand or factor expressions more efficiently. For this problem, we use:

  1. Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
  2. Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

These identities provide a structured way to evaluate expressions that involve cubes.

  1. Simplifying 8^3 - 2^3

To simplify \(8^3 - 2^3\), we identify \(a = 8\) and \(b = 2\). Applying the difference of cubes identity:
\[8^3 - 2^3 = (8 - 2)(8^2 + 8 \cdot 2 + 2^2)\]
Now, we perform the arithmetic inside the parentheses:
\[(8 - 2) = 6\]
\[(8^2 + 8 \cdot 2 + 2^2) = (64 + 16 + 4) = 84\]
Multiplying these results:
\[6 \times 84 = 504\]
Thus, \(8^3 - 2^3 = 504\).

  1. Simplifying 10^3 + 5^3

To simplify \(10^3 + 5^3\), we identify \(a = 10\) and \(b = 5\). Applying the sum of cubes identity:
\[10^3 + 5^3 = (10 + 5)(10^2 - 10 \cdot 5 + 5^2)\]
Now, we perform the arithmetic inside the parentheses:
\[(10 + 5) = 15\]
\[(10^2 - 10 \cdot 5 + 5^2) = (100 - 50 + 25) = 75\]
Multiplying these results:
\[15 \times 75 = 1125\]
Thus, \(10^3 + 5^3 = 1125\).

Examples

Polynomial identities like the sum and difference of cubes are incredibly useful in engineering and physics, where complex expressions often arise in stress analysis or thermal expansion calculations. By factoring or simplifying these expressions using identities, engineers can reduce computational errors and better understand the underlying physical behavior of structures, such as how parts of a machine expand when exposed to high heat. Mastering these identities provides a foundation for more advanced algebraic manipulation in various technical fields.