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 \(n\) has exactly \(n\) roots (counting multiplicities) in the complex number system, which includes both real and imaginary roots.

In this case, the polynomial is \(g(x) = 12x - 3x^2 + 13x^3 - 9\). First, we need to determine the degree of the polynomial. The highest power of \(x\) in \(g(x)\) is \(3\) (from the term \(13x^3\)), so this is a third-degree polynomial.

According to the Fundamental Theorem of Algebra, this polynomial will have \(3\) roots in total (including real and complex roots).

You mentioned that the polynomial has one x-intercept, which means that it has one real root. Since the total number of roots (counting multiplicities) is \(3\) and we already have \(1\) real root, the remaining roots must be complex.

The remaining roots must be either both imaginary (which are non-real) or one of them could be a repeated real root. However, since complex roots must occur in conjugate pairs, the only possibility with \(1\) real root is that the remaining \(2\) roots are non-real.

Therefore, the polynomial will have two imaginary roots.

So the final answer is: two imaginary roots.