Question

What is a polynomial with a single root at x = -1 and a double root at x = 6?

Answer 1

In article <[email protected]>,
[email protected] (Hauke Reddmann) wrote:

> Let a*x**3+b*x**2+c*x+d=0 have a double and a single root.
> What's the value of the single root?

2 3
d (-9 d a + 4 d b c - c )
----------------------------------
2 2 2 2 4
6 d c a + 4 d b - 5 b c d + c

or any other expression equivalent under

2 2 3 2 2 3
(27 d a + 4 b d - 18 c a d b - b c + 4 c a) = 0

which is the discriminant condition for the double root to occur (a being
nonzero assumed).

Maple verification:

> f:=a*x^3+b*x^2+c*x+d;
3 2
f := a x + b x + c x + d

> dis:=factor(resultant(f,diff(f,x),x));
2 2 3 2 2 3
dis := a (27 d a + 4 b d - 18 c a d b - b c + 4 c a)

>
factor(subs(x=d*(-9*d^2*a+4*d*b*c-c^3)/(6*d^2*c*a+4*d^2*b^2-5*b*c^2*d+c^4),f));

3 4 2 3 2 3 3 3 2 2 2 4
d (-27 d a + 36 d a b c - 9 d a c + 16 d b - 24 d b c + 9 d b c

6 2 2 3 2 2 3 /
- c ) (27 d a + 4 b d - 18 c a d b - b c + 4 c a) /
/

2 2 2 2 4 3
(6 d c a + 4 d b - 5 b c d + c )


> (Bonus points if it's possible to write the answer without
> using fractions, just as polynome(a,b,c,d) )

Impossible, multiplying all a,b,c,d by the same factor lambda should
preserve the solution but cannot preserve a polynomial expression.
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