"number of roots of a polynomial equals the number of Sign changes"?

Question

The "number of roots of a polynomial equals the number of Sign changes". Provide with a 3rd order polynomial example,explanation or Proof for this deduction/ conclusion and is this always TRUE? How/ Why ?

Answer 1

Not stated precisely enough. sign changes of what and under what conditions?

Answer 2

Maybe You may be asking about Descarte's Method.

Consider a polynomial f(x)

Then count the number of sign changes.

It will be clear with an example

Let f(x) be 6x^3+4x^2-34x+5

So there are in total 2 sign changes one at 34 and other at 5 .

So what we can conclude is that there are either 2 positive real roots or 0 positive real roots

If the number of sign changes are odd, say 3, then there are either 3 positive real roots or 1 real root.

SO in general if the number of sign changes is even say n , then there are n or n-2 or n-4 .... uptill 0 positive real roots and if it is odd say n , then there are n or n-2 or n-3 ...1 positive real roots.

Now if we substitute -x in f(x) i.e. f(-x) and then count the number of sign changes following the above rule then instead of positive real roots it will be negative real roots.

Thus we can tell by looking at any polynomial about its roots. Your statement that 'number of roots of a polynomial equals the number of Sign changes' means this, I suppose.

I hope you are clear with this explanation.

Answer 3

Yes. Consider f (x)=(x-1)(x-2)^2

Note that x=2 is counted as two roots and not 1 root.

To have an intuitive explanation, we shld plot e curve. Notice that f (0)<0 and f (1.5)> 0 and there is a root x=1. So the root is like the pt where the curve cross over the x axis.

In the case of x=2, 2 there are two roots f (1.5)> 0 and f (2.5)> 0. The sign "changed twice".

Answer 4

If you write f(x)=0. This is a polynomial equation. Then when we plot f(x) vs x , it should cross f(x)=0 at some value of x. If it is a polynomial of degree 2, then it has to cross x axis twice. In the case of third order polynomial, it has to cross three times. But there can be two roots which are complex. One root will be conjugate of the other. In this case, it will cross x axis only once. But if you plot f(x) =0 and see that it will have two kink points. If you draw a line at a suitable value say f(x) = 5, then you will find that f(x) will cross x axis at y=65 line three times.



If f(x) =0 is polynomial order n then it will cross x axis n times.