A 54th-quadraticano polynomial!
P ( x ) = [ ( x 2 + 2 x − 3 ) 3 − ( 4 x 2 + 6 x − 2 ) 3 − ( 3 x 2 + 4 x + 1 ) 3 ] 9
Find the sum of the distinct real roots of P ( x ) .
Give your answer to 3 decimal places.
The answer is -1.333.
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1 solution
You need to show that the 4th degree polynomial has no real roots.
Come on, -4/3 is more accurate than -1.333
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I know. The required answer for the problem is in decimal. I was just following instructions.
But shouldn't the answer be 9 × − 3 4 = 1 2 , since both are repeated roots ?
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I think it means the sum of distinct roots.
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Thank you sir, but i think that should be mentioned in the question.
Let a = x 2 + 2 x − 3 and b = 4 x 2 + 6 x − 2 . We note that b − a = 3 x 2 + 4 x + 1 . Therefore, we have:
P ( x ) = [ ( x 2 + 2 x − 3 ) 3 − ( 4 x 2 + 6 x − 2 ) 3 − ( 3 x 2 + 4 x + 1 ) 3 ] 9 = [ a 3 − b 3 − ( b − a ) 3 ] 9 = [ ( a − b ) ( a 2 + b 2 + a b ) − ( b − a ) 3 ] 9 = [ ( b − a ) ( − a 2 − b 2 − a b − b 2 − a 2 + 2 a b ) ] 9 = [ − ( b − a ) ( 2 a 2 + 2 b 2 − a b ) ] 9 = [ − ( b − a ) ( a 2 + 4 7 b 2 + a 2 − a b + 4 1 b 2 ) ] 9 = [ − ( b − a ) ( a 2 + 4 7 b 2 + ( a − 2 1 b ) 2 ) ] 9
Since a and b have different roots, a 2 + 4 7 b 2 + ( a − 2 1 b ) 2 > 0 and it has no real root, the real roots of P ( x ) is from:
b − a 3 x 2 + 4 x + 1 ( 3 x + 1 ) ( x + 1 ) x = 0 = 0 = 0 = { − 3 1 − 1
Therefore, the sum of the real roots is − 1 − 3 1 ≈ − 1 . 3 3 3