Question

Dennis, Emily, and Fernando want to find the zeros of the polynomial p(x)=x3+6x2+9x. Each student worked independently and presented his or her results to the group. Dennis used the Rational Root Theorem to identify all zeros of p(x). According to this theorem he stated that the zeros of p(x) are ±1, ±3, and ±9. Emily used the greatest common factor and perfect-square trinomial methods to factor p(x) as p(x)=x(x+3)2. She applied the definition of zeros and found that the zeros of p(x) are 0 and −3. Fernando used the greatest common factor and perfect-square trinomial methods to factor p(x) as p(x)=x(x+3)2. He applied the definition of zeros and found that the zeros of p(x) are 0 and −3. He then applied the Irrational Root Theorem and stated that since −3 is a zero, 3 must also be a zero. Therefore, the zeros are 0 and ±3. Which statements accurately justify why each student is correct or incorrect? There may be more than one correct answer. Select all correct answers. emily’s response is incorrect because although she correctly factored the polynomial and used the definition of zeros, she did not apply the Irrational Root Theorem. Fernando’s response is correct because he correctly factored the polynomial, correctly used the definition of zeros, and correctly used the Irrational Root Theorem to identify all zeros. Fernando’s response is incorrect because he inappropriately applied the Irrational Root Theorem. Dennis’ response is incorrect. According to the Fundamental Theorem of Algebra, the polynomial p(x) cannot have six roots, or zeros, because it is only of degree 3. Emily’s response is correct because she correctly factored the polynomial, and correctly used the definition of zeros to reach her answer. Dennis’ response is correct because he correctly factored the polynomial, and correctly used the Rational Root Theorem to identify zeros of a polynomial.

Answer 1

Answer:

• Fernando’s response is incorrect because he inappropriately applied the Rational Root Theorem.
• Dennis’ response is incorrect. According to the Fundamental Theorem of Algebra, the polynomial p(x) cannot have six roots, or zeros, because it is only of degree 3.
• Emily’s response is correct because she correctly factored the polynomial, and correctly used the definition of zeros to reach her answer.

Step-by-step explanation:

The Rational Root Theorem offers a list of possible rational roots. Each needs to be tested to see if it is an actual rational root. Fernando and Dennis made inappropriate assumptions about what the Rational Root Theorem allowed them to conclude.