2018 AIME II Problems/Problem 6
A real number is chosen randomly and uniformly from the interval . The probability that the roots of the polynomial
are all real can be written in the form , where and are relatively prime positive integers. Find .
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, are all possible rational roots. Upon plugging these roots into the polynomial, and make the polynomial equal 0 and thus, they are roots that we can factor out.
The polynomial becomes:
Since we know and 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.
This simplifies to:
This means that the interval is the "bad" interval. The length of the interval where can be chosen from is 38 units long, while the bad interval is 2 units long. Therefore, the "good" interval is 36 units long.
https://www.youtube.com/watch?v=q2oc7n-n6aA ~Shreyas S
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