Question

Suppose that k, a, b, and c are real numbers, a = 0, and a polynomial function P(x) may be expressed in factored form as (x - k)(ax² + bx + c).

(a) What is the degree of P?
A) 0
B) 1
C) 2
D) 3

(b) What are the possible numbers of distinct real zeros of P?
A) 0
B) 1
C) 2
D) 3

(c) What are the possible numbers of non-real complex zeros of P?
A) 0
B) 1
C) 2
D) 3

Answer 1

The degree of P is 2, the possible number of distinct real zeros is 2, and the possible number of non-real complex zeros is 0.

Step-by-step explanation:

(a) Degree of P: The degree of a polynomial is the highest power of x in the equation. In this case, the degree of P(x) is 2 because of the quadratic term ax². Therefore, the correct answer is C) 2.

(b) Number of distinct real zeros of P: The number of distinct real zeros can be determined by the discriminant in the quadratic formula. The discriminant is b² - 4ac. If the discriminant is positive, there are two distinct real zeros. In this case, the discriminant is 10² - 4(1)(-200) = 4040, which is positive. Therefore, the correct answer is C) 2.

(c) Number of non-real complex zeros of P: The number of non-real complex zeros can be determined by the discriminant as well. If the discriminant is negative, there are two non-real complex zeros. In this case, the discriminant is positive, so there are no non-real complex zeros. Therefore, the correct answer is A) 0.