Gary, heather, and irene want to find the zeros of the polynomial p(x). they evaluate the polynomial for different values and find:
p(−1)=0 p(0)=1 p(2+√3)=0
each student interprets this information separately and presents his or her conclusions to the group.
gary concludes that since p(−1) and p(2+√3) equal 0, 2 zeros of p(x) are −1 and 2+√3. by the irrational root theorem, 2−√3 is also a zero of p(x).
heather concludes that since p(0)=1, 1 zero of p(x) is 1. there isn't enough information to determine any other zeros of p(x).
irene concludes that since p(−1) and p(2+3–√) equal 0, 2 zeros of p(x) are −1 and 2+√3. there isn't enough information to determine any other zeros of p(x).
which statements correctly explain why each student is correct or incorrect? there may be more than one correct answer. select all correct answers.
heather is correct. she correctly applied the definition of zeros of a polynomial to identify 1 as a zero of p(x), and correctly stated that there isn't enough information to determine any other zeros of p(x).
irene is correct. she correctly applied the definition of zeroes to identify −1 and 2+3–√ as zeros, and correctly stated that there isn't enough information to determine any other zeros of p(x).
heather is incorrect. she incorrectly used the definition of zeros and she incorrectly stated that there isn't enough information to determine any other zeros of p(x).
gary is incorrect. he correctly used the definition of zeros to identify −1 and 2+√3 as zeros of p(x); however, he incorrectly used the irrational root theorem.
gary is correct. he correctly applied the definition of zeroes to identify −1 and 2+√3 as zeros. he then correctly used the irrational root theorem to find that 2−3–√ is also a zero of p(x).
irene is incorrect. she correctly applied the definition of zeroes to identify −1 and 2+√3 as zeros; however, she incorrectly stated that there isn't enough information to determine any other zeros of p(x).