Question

How can you identify the no. of imaginary roots of a polynomial function?

example: 2x^2 - 3x + 1
positive roots: 2
negative r: 0
imaginary r: 0 or 2



like that and etc
PLEASE PLEASE ANYONE HELP ME

Answer 1

Do you know about Descartes' Rule of signs? @kmeds16

Answer 2

If you don't then, let me state it for you. Descartes' Rule of Signs says that for any polynomial: \[\bf \sum_{k=0}^{n}a_kx^k, \ a_n \ne 0\], the number of sign changes suggests the number of positive roots the polynomial has. For example: If we have the following polynomial:\[\bf f(x)=x^3-5x^2+2x+3\]Then it has at most 2 positive roots because the "signs" from term to term only change twice. The first term has a positive coefficient, the second is negative so that's "1 sign change". Now the third term becomes positive, so that's another sign change. But the last term remains positive so there is no sign change there. Hence we have at most 2 positive roots. But there is a possibility that they are in fact complex roots.

And if they are complex roots then both roots must be complex. This is because complex roots always come in pairs, i.e if \(\bf z=a+bi\) is a complex root, then its conjugate \(\bf z_2=a-bi\) must also be a root of the same polynomial. Hence you can have 0 complex roots, 2 complex roots, 4...so on. You can't have an odd number of complex roots however.

So in this case we counted that the polynomial has 2 changes hence it has at most 2 positive roots. But there is a possibility that the roots are complex and if that's the case, then there can be either 0 complex roots or 2 complex roots. Hence, the polynomial \(\bf f(x)=x^3-5x^2+2x+3\) has either 2 positive roots and 0 complex roots or 2 complex roots and 0 positive roots.

Note that if you'd like to know the number of negative roots of the same polynomial then find \(\bf f(-x)\) and once again count the number of sign changes.

Answer 3

@kmeds16

Answer 4

So for the polynomial you have:\[\bf f(x)=2x^2-3x+1\]We notice that once again we have 2 sign changes. Hence there is either 2 positive roots and 0 complex roots or 2 complex roots and 0 positive roots.

Answer 5

There also follows a rather simple induction proof for the Descartes' Rule of Signs which you can look in to if you wish.

@kmeds16 Do you understand everything I told you so far?

Answer 6

i just saw your reply haha well thanks btw, gonna read it first. yah when i left openstudy i checked out video tutorials about descartes law of signs

Answer 7

@genius12

Answer 8

Ok so do you understand? lol @kmeds16

Answer 9

yeahhh :)