What is difference between number of roots of a polynomial equation and number of solutions to a polynomial equation?
Do they mean same thing OR can an equation have different number of roots and different number of solutions?
They are the same...
Unless they specify, in terms of solutions, that they need "real solutions," or "nontrivial solutions," etc. The roots are always the roots, but what is or is not an acceptable "solution" may depend upon how the question is asked.
EDIT: Also to consider: The above discussion presupposes that by "polynomial equation" you mean one that is given in the format ."
In my precalc book, if you have a polynomial function f(x), the following are equivalent:
* a root of the polynomial f(x),
* a solution to the polynomial equation f(x) = 0,
* a zero of the polynomial f(x), and
* an x-intercept of the polynomial f(x) (assuming that we're only talking about real roots).
Read the first few lines of this:
Roots or zeros of a polynomial - Topics in precalculus
One other consideration comes to mind here. Sometimes roots are counted with multiplicity, and sometimes without. So for instance
has 5 roots counting multiplicity, and 1 root, not counting multiplicity. In either case, there's really only 1 "solution."
Here is just a constant. There will be distinct roots, and therefore "solutions." But if we include multiplicity, there will be roots.
So in general, the amount of "solutions" is less than or equal to the amount of roots, depending on whether or not we count repeated roots.
I hope that answers your question.