Question

If 2 is a polynomial root, name another root of the polynomial, and explain how you know it μst also be a root. (2 points)

a) x=−2, by the factor theorem
b) x=0, by the remainder theorem
c) x=4, by synthetic division
d) x=2, by the fundamental theorem of algebra

Answer 1

Since 2 is a given root of a polynomial, the factor theorem ensures that (x-2) is a factor of the polynomial. The complex conjugate root theorem may apply for real coefficients, but the options do not indicate a complex conjugate pair. Therefore, we cannot name another root without further information.

Step-by-step explanation:

If 2 is a root of a polynomial, then according to the factor theorem, x-2 is a factor of the polynomial. The factor theorem states that if x-a is a factor of the polynomial, then a is a root, and vice versa. Therefore, if 2 is a root, another root could possibly be the complex conjugate if the polynomial has real coefficients. However, none of the options provided suggest a complex conjugate. Without additional information about the polynomial or its coefficients, we cannot definitively name another root based on the information given.