Questions: Which of the following endpoints is best suited for using polynomial identities to convert differences of numerical squares into products? (1 point) 8 and 10 7.2 and 10 8.4 and 11.1 8 and 11.5

Which of the following endpoints is best suited for using polynomial identities to convert differences of numerical squares into products? (1 point)
8 and 10
7.2 and 10
8.4 and 11.1
8 and 11.5

Transcript text: Which of the following endpoints is best suited for using polynomial identities to convert differences of numerical squares into products? (1 point) 8 and 10 7.2 and 10 8.4 and 11.1 8 and 11.5

Solution

Solution Steps

To determine which pair of endpoints is best suited for using polynomial identities to convert differences of numerical squares into products, we can use the identity \(a^2 - b^2 = (a-b)(a+b)\). We will calculate the difference and sum for each pair and check which pair results in integers or simple fractions.

Step 1: Identify the Polynomial Identity

We use the identity \(a^2 - b^2 = (a-b)(a+b)\) to convert differences of squares into products. We need to find which pair of endpoints results in simple integer or fractional values for \(a-b\) and \(a+b\).

Step 2: Calculate Differences and Sums

For each pair of endpoints, calculate \(a-b\) and \(a+b\):

For \((8, 10)\):
- \(a-b = 8 - 10 = -2\)
- \(a+b = 8 + 10 = 18\)
For \((7.2, 10)\):
- \(a-b = 7.2 - 10 = -2.8\)
- \(a+b = 7.2 + 10 = 17.2\)
For \((8.4, 11.1)\):
- \(a-b = 8.4 - 11.1 = -2.7\)
- \(a+b = 8.4 + 11.1 = 19.5\)
For \((8, 11.5)\):
- \(a-b = 8 - 11.5 = -3.5\)
- \(a+b = 8 + 11.5 = 19.5\)

Step 3: Determine the Best Pair

The best pair is the one where both \(a-b\) and \(a+b\) are integers or simple fractions. The pair \((8, 10)\) results in integer values for both \(a-b\) and \(a+b\).