Question

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)
Responses

no imaginary roots
no imaginary roots

one imaginary root
one imaginary root

three imaginary roots
three imaginary roots

two imaginary roots

Answer 1

To apply the Fundamental Theorem of Algebra, we first note that it states that a polynomial of degree \( n \) will have exactly \( n \) roots in the complex number system, counting multiplicities.

The polynomial given is:

\[
g(x) = 12x - 3x^2 + 13x^3 - 9
\]

To determine the degree of the polynomial, we look at the highest power of \( x \), which is \( 3 \) (from the term \( 13x^3 \)). Thus, the polynomial is of degree \( 3 \).

According to the Fundamental Theorem of Algebra, this means \( g(x) \) will have exactly \( 3 \) roots in total (some of which may be real, and some may be imaginary).

We are informed that the polynomial has one x-intercept, implying that there is one real root. The remaining roots must then account for the total of 3 roots.

Since polynomials with real coefficients that do not have complex roots come in conjugate pairs, the remaining \( 2 \) roots must be complex. Thus, there are \( 2 \) imaginary roots.

So, the answer is:

**two imaginary roots**.