a real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. the probability that the roots of the polynomial $x^4 2ax^3 (2a - 2)x^2 (-4a 3)x - 2$ are all real can be written in the form $\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. find $m n$.

The polynomial we are given is rather complicated, so we could use Rational Root Theorem to turn the given polynomial into a degree-2 polynomial. With Rational Root Theorem, x = 1, -1, 2, -2 are all possible rational roots. Upon plugging these roots into the polynomial, x = -2 and x = 1 make the polynomial equal 0 and thus, they are roots that we can factor out.

The polynomial becomes:

(x - 1)(x + 2)(x^2 + (2a - 1)x + 1)

Since we know 1 and -2 are real numbers, we only need to focus on the quadratic.

We should set the discriminant of the quadratic greater than or equal to 0.

(2a - 1)^2 - 4 ≥ 0.

This simplifies to:

a ≥ 3/2

or,

a ≤ - 1/2

This means that the interval (- 1/2 ,3/2) is the "bad" interval. The length of the interval where a can be chosen from is 38 units long, while the bad interval is 2 units long. Therefore, the "good" interval is 36 units long.

36/38 = 18/19

18+ 19 = 037

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