Question

Apply the Fundamental Theorem of Algebra to determine how many imaginary roots g(x)=12x−3x2+13x3−9
𝑔
(
𝑥
)
=
12
𝑥
−
3
𝑥
2
+
13
𝑥
3
−
9
will have, if you know it has one x-intercept.(1 point)
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 exactly \( n \) roots in the complex number system, counting multiplicities.

The polynomial given is:
\[
g(x) = 12x - 3x^2 + 13x^3 - 9
\]
This polynomial is a cubic polynomial (degree 3), which means it will have a total of 3 roots (real or complex).

You mentioned that it has one x-intercept, which means there is one real root. The remaining roots can either be real roots or imaginary roots. Since we already have one real root, the remaining two roots must be accounted for.

If there is one real root, the other two roots must be complex, forming a complex conjugate pair (because the coefficients of the polynomial are real). Therefore, there will be two imaginary roots.

Thus, the answer is:
**two imaginary roots**.