The best way to identify which pairs of these numerical squares can be converted into products using polynomial identities is to recognize that the difference of squares is represented by the formula \( a^2 - b^2 = (a - b)(a + b) \).
To apply this to your endpoints:
-
For the pair \(8\) and \(11.5\):
- Difference: \( 11.5^2 - 8^2 \)
-
For the pair \(7.2\) and \(10\):
- Difference: \( 10^2 - 7.2^2 \)
-
For the pair \(8.4\) and \(11.1\):
- Difference: \( 11.1^2 - 8.4^2 \)
-
For the pair \(8\) and \(10\):
- Difference: \( 10^2 - 8^2 \)
To determine the most suitable pair for converting differences of squares into products, we would calculate the numerical values of these differences, if needed, but in most cases, the choice can be made by checking the numerical calculations or deciding based on a simple difference approach.
Let's check each option:
\(8\) and \(10\) are easily recognized as being whole numbers, and the calculation yields a simple result.
If you want to pursue numerical values:
- \( 11.5^2 - 8^2 \) = \(132.25 - 64\) = \(68.25\)
- \( 10^2 - 7.2^2 \) = \(100 - 51.84\) = \(48.16\)
- \( 11.1^2 - 8.4^2 \) = \(123.21 - 70.56\) = \(52.65\)
- \( 10^2 - 8^2 = 100 - 64 = 36\)
Based on simplicity and the fact that these differences yield easily factorable results, landing on the pairs of simple whole numbers related to \(8\) and \(10\) is usually most efficient.
Thus, the pair 8 and 10 would be considered the best suited for this purpose, as they yield a simple, manageable difference and conform neatly to the \( a^2 - b^2 \) identity transformation.