Question

The leading term of the polynomial function p is a_nx^n , where a_n is a real number and 1 is a positive integer. The factors of p include (x-3),(x-i) , and (x-(2+i)). What is the least possible value of n ? a. 3 b. 1 c. 5 d. 6

Answer 1

The least possible value of n is 5, considering the requirement for complex conjugate pairs in a polynomial with real coefficients, which gives us three distinct factors that lead to a fifth-degree polynomial.

Step-by-step explanation:

The least possible value of n is 3, which corresponds to option a. This is based on the minimum requirement for complex conjugate pairs and distinct linear factors in a polynomial. As given, the factors of the polynomial p include (x-3), (x-i), and (x-(2+i)). Because polynomials with real coefficients must have complex roots in conjugate pairs, the factor (x-i) necessitates the conjugate factor (x+i), and similarly (x-(2+i)) necessitates the conjugate factor (x-(2-i)).

Therefore, we have three distinct factors when considering the conjugate pairs: (x-3), (x2+1), and (x2+4x+5), giving us a minimum degree of 1+2+2=5. Since the problem specifies that an is a real number, and n is a positive integer, we can confirm that option c with n=5 is the correct answer.