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

two imaginary roots
two imaginary roots

one imaginary root
one imaginary root

three imaginary roots
three imaginary roots

no imaginary roots

Answer 1

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

For the polynomial \( g(x) = 12x - 3x^2 + 13x^3 - 9 \), we first need to determine the degree of the polynomial. The highest degree term is \( 13x^3 \), indicating that this polynomial is of degree 3.

Since a cubic polynomial (degree 3) has three roots in total, and you mentioned that it has one x-intercept (which corresponds to one real root), we can determine the number of imaginary roots by subtracting the number of real roots from the total roots:

• Total roots (from the Fundamental Theorem of Algebra): 3 (since it is cubic)
• Real roots: 1 (the given x-intercept)

This means the number of imaginary roots is:

\[ 3 , \text{(total roots)} - 1 , \text{(real root)} = 2 , \text{(imaginary roots)} \]

Thus, the answer is:

two imaginary roots.