Question
Which of the following endpoints is best suited for using polynomial identities to convert differences of numerical squares into products?(1 point)Responses8.4 and 11.18.4 and 11.18 and 108 and 108 and 11.58 and 11.57.2 and 10 pick one
Answers
GPT-4o mini
The best suited endpoints for using polynomial identities to convert differences of numerical squares into products would be the ones that can be expressed as a difference of squares, which follows the identity:
\[
a^2 - b^2 = (a - b)(a + b)
\]
Looking at the options provided, we need to calculate the differences for each pair:
1. **8.4 and 11.1:** \(11.1^2 - 8.4^2\)
2. **8.4 and 11.1:** \(11.1^2 - 8.4^2\)
3. **8 and 10:** \(10^2 - 8^2\)
4. **8 and 11.5:** \(11.5^2 - 8^2\)
5. **8 and 11.5:** \(11.5^2 - 8^2\)
6. **7.2 and 10:** \(10^2 - 7.2^2\)
However, we are looking for integer values (or simple fractions that can clearly be expressed with integers). The differences of squares can be calculated for each of the pairs provided to check which one provides an easy approach for substituting into the difference of squares formula.
From these calculations, the pairs that yield the easiest products based on the differences of squares would typically involve integers rather than decimals.
Thus, the least complicated numbers that maximize this are:
**8 and 10:**
This yields the calculation:
\[
10^2 - 8^2 = (10 - 8)(10 + 8) = 2 \cdot 18 = 36
\]
So, I would select **8 and 10** as the endpoints best suited for using polynomial identities to convert differences of numerical squares into products.
\[
a^2 - b^2 = (a - b)(a + b)
\]
Looking at the options provided, we need to calculate the differences for each pair:
1. **8.4 and 11.1:** \(11.1^2 - 8.4^2\)
2. **8.4 and 11.1:** \(11.1^2 - 8.4^2\)
3. **8 and 10:** \(10^2 - 8^2\)
4. **8 and 11.5:** \(11.5^2 - 8^2\)
5. **8 and 11.5:** \(11.5^2 - 8^2\)
6. **7.2 and 10:** \(10^2 - 7.2^2\)
However, we are looking for integer values (or simple fractions that can clearly be expressed with integers). The differences of squares can be calculated for each of the pairs provided to check which one provides an easy approach for substituting into the difference of squares formula.
From these calculations, the pairs that yield the easiest products based on the differences of squares would typically involve integers rather than decimals.
Thus, the least complicated numbers that maximize this are:
**8 and 10:**
This yields the calculation:
\[
10^2 - 8^2 = (10 - 8)(10 + 8) = 2 \cdot 18 = 36
\]
So, I would select **8 and 10** as the endpoints best suited for using polynomial identities to convert differences of numerical squares into products.