apply the fundamental theorem of algebra to determine how many imaginary roots g(x) = 12x - 3x^2 + 13x^3 - 9 will have, if you know it has one x-intercept
two imaginary roots
no imaginary roots
one imaginary root
three imaginary roots
The fundamental theorem of algebra states that a polynomial of degree n will have exactly n complex roots, counting multiplicities. In this case, the polynomial g(x) = 12x - 3x^2 + 13x^3 - 9 has a degree of 3 (the highest power of x is 3).
If we know that g(x) has one x-intercept, it means that it has one real root (also known as a zero). The real root represents one of the complex roots.
Since g(x) has a degree of 3 and we know it has one real root, by the fundamental theorem of algebra, it must have 3 total roots. Therefore, it will have two additional complex (imaginary) roots.
Therefore, the answer is:
Two imaginary roots.