Find all real m such that x 3 − 2 x 2 − 2 x + m has 3 distinct rational roots.

Question

Find all real m such that       x    3    −  2      x    2    −  2  x  +  m has 3 distinct rational roots.

Kaitlynn Church · Accepted Answer

Let a,b,c be the rational roots. Then we have

a + b + c = 2, a b + b c + c a = - 2, m = - a b c .

Replace

c = 2 - a - b

a^{2} + b^{2} + a b - 2 (a + b) - 2 = 0.

(If a=b, then a is irrational) Thus,

Δ_{b} = (a - 2)^{2} - 4 (a^{2} - 2 a - 2) = - 3 a^{2} + 4 a + 12

is a square rational.
The rest is easy.
Added after diner. Since

- 3 a^{2} + 4 a + 12 = - 3 (a + \frac{2}{3})^{2} + 4 \times \frac{10}{3}

is a square. Let

\frac{3}{2} (a + \frac{2}{3}) = \frac{p}{q}

(suppose

gcd (p, q) = 1

), then

- 3 p^{2} + 30 q^{2}

is a square of integer, say 3r, thus

- p^{2} + 10 q^{2} = 3 r^{2} .

It is clear that

2 | p - r

. If both are even, then q is even too, contradicting

gcd (p, q) = 1

. So p and r both are odd, then

p^{2} + 3 r^{2} = 4 \mod (8)

, therefore q is even, hence

10 q^{2} = 0 \mod (8)

, this is an contradiction with

p^{2} + 3 r^{2} = 10 q^{2}

Find all real m such that x^3−2x^2−2x+m has 3 distinct rational roots.