Question
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
two imaginary roots
no imaginary roots
one imaginary root
three imaginary roots
Answers
GPT 3.5
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.
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.