Polynomial 2.2
x 3 + 2 x 2 − x = 2
What is the sum of all the solutions to the equation above?
The answer is -2.
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3 solutions
Abhay where does this come from?
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Vieta's formula, click the link in his solution to learn more
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If I understood it the equation -b/a applies to second degree equations? This is a third degree equation.
Click Vieta's formula link and learn it.
Ah, I totally forgot I could do this instead!
x 3 + 2 x 2 − x − 2 = 0
Notice that: 1 + 2 − 1 − 2 = 0
Therefore, x = 1 is a solution to the equation.
Rewrite:
x 3 − x 2 + 3 x 2 − 3 x + 2 x − 2 = 0 x 2 ( x − 1 ) + 3 x ( x − 1 ) + 2 ( x − 1 ) = 0 ( x − 1 ) ( x 2 + 3 x + 2 ) = 0 ( x − 1 ) ( x + 2 ) ( x + 1 ) = 0 x = − 2 , − 1 , 1
The sum of all the solutions = − 2 + ( − 1 ) + 1 = − 2
x 3 + 2 x 2 − x − 2 = 0
x 2 ( x + 2 ) − ( x + 2 ) = 0
( x + 2 ) [ x 2 − 1 ] = 0
( x + 2 ) ( x − 1 ) ( x + 1 ) = 0
From the last equation you can see three equations
x + 2 = 0 , x = − 2
x − 1 = 0 , x = 1
x + 1 = 0 , x = − 1
The sum of the three solutions is (-2) + 1 + (-1) = -2