How can I calculate Total no. of real roots in
Last edited by jacks (2013-05-20 21:13:16)
I can compute all the roots you need but keep in mind as an answer to your question we know everything about polynomials and very little about everything else. Except for trick questions which make there way into books there are no general methods. CAS combine dozens of methods and iterative searches to get all the roots.
In other words to know how many we have to compute them and count.
1) x = 0 and x = 1 are obviously roots.
Plotting is the first move for any general attack on an equation. This shows tentatively there is one more root between 4 and 5. See the middle picture.
To prove that there are no more we maybe can use calculus here.
2) For this one there looks like there is one root near - 1 / 2. ( see the last image )
3) There looks like there is one root around - 3 / 5. ( see the first image )
As a general rule when an exponential function begins to overpower a polynomial, the polynomial never recovers. I would say that there are no more roots.
In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.