Oops, is it lengthy?
Find sum of negative roots of the following:
x 4 + 9 x 3 + 1 2 x 2 − 8 0 x − 1 9 2 = 0
Count repeated roots with multiplicity.
The answer is -12.
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1 solution
Well you can't directly say that, because there might be some non-real roots at times. So first we need to check if all the roots are real or not, which, in this case, are all real.
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(Check for negative roots) By changing signs of odd powers, you get three sign changes., indicating three negative roots. Observation is enough. Calculation is not required.
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No No No!!! When you get 3 sign changes, it implies there are either 3 or 1 negative roots.
This is like if you get 10 sign changes, it means there are 10 or 8 or 6 or 4 or 2 or 0 roots of the type. Thus if there are odd number of sign changes, it means there has to be AT LEAST ONE root of needed type. Note this, it makes a big difference!!
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@Aditya Raut – Well said. That's a very common mistake made with Descartes Rule of Signs .
@Aditya Raut – Good that you corrected me. I am often wrong in over simplifying.
A simple example to demonstrate, see
f ( x ) = x 4 + x 3 − x − 1 = 0
Now number of sign changes for f ( x ) are 1 . Hence it has 1 positive root.
But f ( − x ) = x 4 − x 3 + x − 1 this has 3 sign changes.
But you know what? There are not 3 negative roots, there's just one negative root.
x 4 + x 3 − x − 1 = 0 ∴ x 3 ( x + 1 ) − 1 ( x + 1 ) = 0 ∴ ( x + 1 ) ( x 3 − 1 ) = 0 ∴ ( x + 1 ) ( x − 1 ) ( x 2 + x + 1 ) = 0
And x 2 + x + 1 = 0 has got no real roots, so you can't define +ve and -ve for them, I think I have made it quite clear now....
according to me
x 4 + 9 x 3 + 1 2 x 2 − 8 0 x − 1 9 2 = ( x − 3 ) ( x + 4 ) 3 so there are three negative roots ( − 4 , − 4 , − 4 ) and their sum is − 1 2
what say @Mehul Chaturvedi
Dear @Calvin Lin Sir, I feel that there is just one negative root, − 4 , so why do we sum it 3 times? I think I am missing something. Please help me. Thanks!
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I added "Count repeated roots with multiplicity. ".
As change of signs is one there is only one positive root and that is 3 (Found by inspection), As the sum of all the roots is 9, the remaining three negative roots must add up to 12.