I was solving the problem listed here..
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?
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
the original question deals with a field of characteristic p, so a real-valued plot is not especially helpful, here (since char(R) = 0).
Thanks for your help DEVENO
i presume the logic to be this.
since r is a root, r^4 + r + 6 = 0, and 4r^3 + 1 = 0 so, multiplying any constant/variable other than 4 and r will also return a zero. but choosing 'r' and '4' helps us because, we are able to get rid of r^4 helping in a simpler form of a polynomial.
i was able to follow after that...
That was of great help.