SOLUTION: I am trying to help a student at a residential treatment center. We are looking for the number and type of complex solutions and possible real solutions for the following: 2x^2

Question 564982: I am trying to help a student at a residential treatment center. We are looking for the number and type of complex solutions and possible real solutions for the following:
2x^2+5x+3=0
4x^3-12x+9=0
2x^4+x^2-x+6=0
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
The number of complex solutions of a polynomial is always equal to the degree of the polynomial (this is called the fundamental theorem of algebra).

You can find the possible rational roots quite easily using the rational root theorem. The possible rational roots of a polynomial are in the form

where p is a factor of the constant term and q is a factor of the leading coefficient. This theorem can easily be proven using modular arithmetic.

There isn't much of a way to find possible "real" roots. However, you do know that if P(a) is negative and P(b) is positive, there exists at least one real zero between a and b. This is due to the intermediate value theorem, which states that for a continuous function f(x) between a and b, every number between f(a) and f(b) has at least one x-value in the domain.
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