Apply the Fundamental Theorem of Algebra to determine how many imaginary roots g(x)=12x−3x2+13x3−9 will have, if you know it has one x-intercept.(1 point)

Question

Responses

three imaginary roots
three imaginary roots

one imaginary root
one imaginary root

two imaginary roots
two imaginary roots

no imaginary roots

Answer 1

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) will have exactly \( n \) roots in the complex number system (which includes both real and imaginary roots).

The polynomial \( g(x) = 12x - 3x^2 + 13x^3 - 9 \) is a cubic polynomial, which means its degree \( n \) is 3. Therefore, it will have exactly 3 roots, counting multiplicities.

You mentioned that the polynomial has one x-intercept. Since the x-intercept corresponds to a real root, we can conclude the following:

• If there is one real root, then the other roots must be either real or a pair of complex conjugate roots (which are considered imaginary).
• In the case of a cubic polynomial, the possible combinations of roots considering that one root is real are:
  • 1 real root and 2 non-real roots (which are imaginary).

Thus, if the polynomial has one x-intercept (i.e., one real root), it will have two imaginary roots.

The answer is: two imaginary roots.