how to know of the number of real roots? Let $ax^4 +bx^3 +cx^3 +dx + e = 0$ with $a,b,c,d,e\in\mathbb R$. I would like to know, how can I determine the condition for the polynomial to have exactly three distinct real solutions.
one has to be a double root, or is there any other possibility.
I need help please.
Thanks
 A: Hint: The complex conjugate root theorem (for polynomials with real coefficients) states that complex roots occur in conjugate pairs.
Hint: If we have 3 distinct real roots to a degree 4 equations, how many complex roots do we have?
A: Depending what you mean, if the equation has a double root eg $x^2(x+1)(x-1)=0$ it can have three distinct real roots, but we've counted one twice.
So there are two conditions:
(i) One double root
(ii) The other two roots are real and distinct
A: Usually roots are counted according to multiplicity of roots. So, in my humble opinion, it is assumed that first coefficient is zero: $a=0$. Thus we get cubic equation and there are conditions on discriminant for cubic equation such that it has 3 roots. See, for instance, http://en.wikipedia.org/wiki/Cubic_function
A: Complex roots(in a polynomial with real coefficients) occur in pairs so the no. of real roots can be either $2$ or $4$
If there are 3 distinct real roots then the no. of real root(not necessarily distinct) is 4 hence there are no complex root.
