Question

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?

Answer 1

The answer is given below

Explanation:

A number is said to be a zero of a polynomial if when the number is substituted into the function the result is zero. That is if a is a zero of polynomial f(x), therefore f(a) = 0.

Since P(−1)=0 P(0)=1 P(2+√3)=0, therefore -1 and 2+√3 are zeros of the polynomial.

Gary is right because there are 2 known zeros of P(x) which are −1 and 2+√3. Also 2 - √3 is also a root. From irrational root theorem, irrational roots are in conjugate pairs i.e. if a+√b is a root, a-√b is also a root.

Heather is not correct because if P(0) = 1, it means that 0 is not a root. It does not mean that 1 is a zero of P(x)

Irene is correct. since P(−1) and P(2+3–√) equal 0, 2 zeros of P(x) are −1 and 2+√3. They may be other zeros of P(x), but there isn't enough information to determine any other zeros of P(x)