A Polynomial and a Triangle
Let
be a polynomial with
positive roots.
A triangle with side lengths equal to the roots of
has an area of
Find .
The answer is 0.5.
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1 solution
Note that the sum of the roots is 4.
Now, if the roots are a, b, and c, then the area of the triangle is ( 2 a + b + c ) ( 2 a + b + c − a ) ( 2 a + b + c − b ) ( 2 a + b + c − c ) by Heron's formula.
But we realize that since f ( x ) factors into ( x − a ) ( x − b ) ( x − c ) , the area is just ( 2 a + b + c ) f ( 2 a + b + c ) !
Thus ( 2 ) f ( 2 ) = 1 → f ( 2 ) = . 5 .