# Thread: roots with multiplicity > 1 polynomials

1. ## roots with multiplicity > 1 polynomials

I was solving the problem listed here..

Polynomial x

I understood the logic behind the first line which says that "A root is of multiplicity 1 ==> the polynomial has a root in common with it's own derivative.

But I wasn't able to follow the next where the reply is like this.
Thus the given information tells us that x^4 + x + 6 and 4x^3 + 1 have a common root r. It follows that
0 = 4(r^4 + r + 6) - r(4r^3 + 1) = 3r + 24.

I could not follow the logic behind the line 0 = 4(r^4 + r + 6) - r(4r^3 + 1) = 3r + 24.

I have been searching for the justification behind this step, but have been failing again and again. Can someone throw some light on this?

Thanks

3. ## Re: roots with multiplicity > 1 polynomials

if r is a root of x4 + x + 6 and 4x3 + 1, then:

r4 + r + 6 = 0, and 4r3 + 1 = 0, hence:

4(r4 + r + 6) - r(4r3 + 1) = 4*0 - r*0 = 0.

expanding, we have:

4r4 + 4r + 24 - 4r4 - r = 0, and the r4 terms cancel out leaving:

3r + 24 = 0

@MaxJasper:

the original question deals with a field of characteristic p, so a real-valued plot is not especially helpful, here (since char(R) = 0).

4. ## Re: roots with multiplicity > 1 polynomials

the graph helped with a visual perspective of the problem.

Thanks, max Jasper