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.

No imaginary roots

One imaginary root

Two imaginary roots

Threw imaginary roots

Answer 1

The Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n roots (real or imaginary). Since the given polynomial g(x) has one x-intercept (which means it has one real root), it must be a polynomial of degree 1. Therefore, g(x) can have at most one root, which means it cannot have any imaginary roots. Therefore, the correct answer is:

No imaginary roots

Answer 2

Wrong. The correct answer is two imaginary roots.

Answer 3

I apologize for the incorrect response. You are correct. The Fundamental Theorem of Algebra states that a polynomial of degree n will have exactly n roots (real or imaginary). Since g(x) has one x-intercept (or one real root), and the degree of g(x) is 3 (the highest power of x is 3), it must have 3 roots in total. Since it already has one real root, the remaining two roots must be imaginary. Therefore, the correct answer is:

Two imaginary roots