To determine which endpoints are best suited for using polynomial identities to convert differences of numerical squares into products, we can refer to the identity:
\[ a^2 - b^2 = (a - b)(a + b) \]
This means that if we have two perfect squares that differ, we can apply this identity effectively.
Let's analyze the options you provided to see if they are suitable for the identity. We will check the endpoints you mentioned one by one:
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8.4 and 11.1:
- Difference: \(11.1 - 8.4 = 2.7\), not a perfect square.
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8.4 and 11.1:
- Already analyzed above.
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8 and 10:
- Difference: \(10 - 8 = 2\), not a perfect square.
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8 and 11.5:
- Difference: \(11.5 - 8 = 3.5\), not a perfect square.
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8 and 11.5:
- Already analyzed above.
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7.2 and 10:
- Difference: \(10 - 7.2 = 2.8\), not a perfect square.
To find pairs where the difference of the squares yields perfect squares, we would ideally want pairs of integers where the difference aligns with some usable value.
Since none of these pairs provided are direct integers or lead to clear differences of perfect squares, none match easily to produce useful outputs using the identity directly.
Given this, if we need to select based on numerical estimates that keep within bounds, we can't use these naive calculations. However, "8 and 11" would serve as common integers for explaining point differences easily in classroom settings; hence there is no explicitly suitable pair listed.
In retrospect, I advise checking exact endpoints or pairs that could yield squares in practical settings.
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