Find multiple roots (repeated roots) of a polynomial depending on the real parameter "a"

Question

Hello everyone! I have got this task: Find multiple roots (repeated roots) of a polynomial \(\displaystyle t^3 - (1-2a)t^2-(a^2-2a)t+a^2\) depending on the real parameter a. My idea was to factorize the polynomial using the Euclidean algorithm, but I was unable to do it , because it's becoming larger and larger every single step and I don't know what to do..Could anyone help me or give me a hint?

Answer 1

suppose the given polynomial factors \(\displaystyle t^3-(1-2a)t^2-\left(a^2-2a\right)t+a^2=(t-b)(t-c)^2\) expanding and equating we get a system of equations \(\displaystyle a^2+b c^2=0\) \(\displaystyle 2 a-a^2-2 b c-c^2=0\) \(\displaystyle -1+2 a+b+2 c=0\) the machine solves this and gives us \(\displaystyle a=c^3\) and \(\displaystyle b=-c^4\) where \(\displaystyle c\) is a root of the polynomial \(\displaystyle 1-2 c-2 c^3+c^4\) additional solution exists when \(\displaystyle c=0, b=1, a=0\)

Answer 2

topsquark said: Addendum: I know you already verified the equation but are you sure that last term isn't a \(\displaystyle a^3\)? Click to expand... Yes, absolutely sure.

Answer 3

The only real way I can see to attack this is to use Calculus. There is at least a double root so the graph must just touch the x-axis so the local max or min is on the x-axis and we can use that to solve for a. To do this with only algebra you would need to solve the cubic for t (use Cardano's method) and adjust the value of a such that you have at least a double root. Either method is going to get very ugly and beyond what I would expect for a High School student. My thought here is to try some values. Start with some integers and see what you get. (I haven't found one but I haven't spent much time with it.) -Dan Addendum: I know you already verified the equation but are you sure that last term isn't a \(\displaystyle a^3\)?

Answer 4

Debsta said: Just checking. Is the second bracket correct? Click to expand... Yes, it's correct

Answer 5

the polynomial and its derivative must have a common root so their resultant must be \(\displaystyle 0\) \(\displaystyle -8 a^2 \left(1-14 a+24 a^2-14 a^3+a^4\right)=0\)

Answer 6

Just checking. Is the second bracket correct?