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{ x 5 − x 3 + x 2 − 1 = 0 x 4 = 1
Let the two common roots of the above equations be a and b .
Find a + b .
The answer is 0.
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3 solutions
x 4 has roots ± 1, ± ι of which only ± 1 satisfy first equation hence ± 1 are common roots of the given equations . Hence, 1 + ( − 1 ) = 0 Another slightly different method is substitute 1= x 4 in the first equation and do factorisation to get x= ± 1
There are two other roots of x 4 .
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Yup ± ι , but they don't satisfy the first equation..
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Yes, you probably want to explicitly mention in the solution that there are those other roots that don't satisfy the first equation (because otherwise people like me above think that you missed the other two roots and didn't check them).
x 5 − x 3 + x 2 − 1 = 0 ⇒ ( x 2 − 1 ) ( x 3 + 1 ) = 0 ⇒ x = 1 or − 1 or 2 1 ± − 3
Substituting these roots into the second equation, we find that only the first two roots are common, hence a = 1 , b = − 1 , a + b = 0 .
Do a factorization. x 5 − x 3 + x 2 − 1 = ( x 2 − 1 ) ( x 3 + 1 ) and x 4 − 1 = ( x 2 − 1 ) ( x 2 + 1 ) . Here, the GCD of them is x 2 − 1 , hence the common root is x = 1 , − 1 and its sum is 1 − 1 = 0