Question

The function Q is a polynomial of degree 3. If Q(5)=0 , which of the following must be true?

a. Q(-5)=0
b. Q has two complex zeros.
c. Q(x) can be expressed as (x-5)· P(x) , where P(x) is a polynomial of degree 2.
d. Q(x) can be expressed as P(x)/x-5 , where P(x) is a polynomial of degree 4

Answer 1

a. Q(-5) = 0 may or may not be true. b. Q does not necessarily have two complex zeros. c. Q can be expressed as (x-5) · P(x), where P(x) is a polynomial of degree 2. d. Q cannot be expressed as P(x)/(x-5), where P(x) is a polynomial of degree 4.

Step-by-step explanation:

Given that the function Q is a polynomial of degree 3 and Q(5) = 0, we can make the following determinations:

1. a. Q(-5) = 0: This statement is not necessarily true. The value of Q(-5) could be non-zero, as it depends on the specific polynomial.
2. b. Q has two complex zeros: This statement is not necessarily true either. A polynomial of degree 3 can have three distinct real zeros, including one at x = 5, or it can have one real zero and two complex conjugate zeros.
3. c. Q(x) can be expressed as (x-5) · P(x), where P(x) is a polynomial of degree 2: This statement is true. If Q(5) = 0, it means that (x-5) is a factor of Q(x).
4. d. Q(x) can be expressed as P(x)/(x-5), where P(x) is a polynomial of degree 4: This statement is not necessarily true. Since Q is a polynomial of degree 3, it cannot be expressed as the quotient of a polynomial of degree 4 and (x-5).