The Fundamental Theorem of Algebra Quick Check

Question

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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

three imaginary roots
three imaginary roots

no imaginary roots
no imaginary roots

one imaginary root
one imaginary root

two imaginary roots

Answer 1

To apply the Fundamental Theorem of Algebra, we need to consider the properties of polynomial functions. The theorem states that a polynomial of degree \( n \) will have exactly \( n \) roots in the complex number system (which includes both real and imaginary roots).

In the function you provided, \( g(x) = 12x - 3x^2 + 13x^3 - 9 \):

1. Identify the degree of the polynomial. The term with the highest degree is \( 13x^3 \), which indicates that this polynomial is of degree 3.
2. According to the Fundamental Theorem of Algebra, since the degree is 3, there are a total of 3 roots (real or complex).
3. You mentioned that there is one \( x \)-intercept, meaning that there is one real root.

Since complex roots come in conjugate pairs, if there is one real root, the remaining roots must be complex.

Given that there are a total of 3 roots and you have identified 1 real root:

• 3 total roots - 1 real root = 2 roots must be complex.

Thus, these 2 remaining roots must be complex and therefore will be imaginary (since they aren't real).

So, the answer is:

two imaginary roots