The side lengths of a scalene triangle are the roots of the polynomial x`^3` - 20x`^2` +131 x -281.3. Find the square of the area of the triangle.

The square of the area of this triangle is 290.

Expert Answers

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The square of the square of a triangle with sides `a , b , c ` is `S ^ 2 = p ( p - a ) ( p - b ) ( p - c ) , ` where `p ` is the semiperimeter, `p = 1 / 2 ( a + b + c ) . ` It is Heron's formula.

From the other hand, our polynomial has the form `f ( x ) = ( x - a ) ( x - b ) ( x - c ) , ` although we don't know `a , b ` and `c .`

If we'd know `p , ` the answer would be `S^2 = p * f ( p ) . ` Actually, it is simple to find it. Use Vieta's formulas for a cubic equation, or open the parentheses in `x^3 + Bx^2 + Cx + D = ( x - a ) ( x - b ) ( x - c ) ` directly and obtain `a + b + c = -B` (coefficient at `x^2 ` ).

This way, `p = -B / 2 = 10 ` and

`S ^ 2 = 10 * f ( 10 ) = 10 * ( 1 00 0 - 20 00 + 13 10 - 2 81 ) = 290 .`

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