Lagrange Interpolation might be useful

Algebra Level 3

Let P P be a cubic monic polynomial with roots a a , b b , and c c . If P ( 1 ) = 91 P(1)=91 and P ( − 1 ) = − 121 P(-1)=-121 , compute the maximum possible value of a b + b c + c a a b c + a + b + c . \dfrac{ab+bc+ca}{abc+a+b+c}.


The answer is 7.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Sharky Kesa
Aug 20, 2017

Let x = a b c + a + b + c x=abc+a+b+c and y = a b + b c + c a + 1 y=ab+bc+ca+1 .

Thus, we have P ( 1 ) = 1 − ( a + b + c ) + ( a b + b c + c a ) − a b c = 91 P(1)=1-(a+b+c)+(ab+bc+ca)-abc=91 by Vieta's, so − x + y = 91 -x+y=91 .

We also have P ( − 1 ) = − 1 − ( a + b + c ) − ( a b + b c + c a ) − a b c = − 121 P(-1) = -1 - (a+b+c) - (ab+bc+ca) - abc = -121 , so x + y = 121 x+y=121 .

Thus, we have x = 15 x=15 and y = 106 y=106 . Therefore, a b + b c + c a a b c + a + b + c = y − 1 x = 105 15 = 7 \dfrac{ab+bc+ca}{abc+a+b+c}=\dfrac{y-1}{x} = \dfrac{105}{15}=7 .

1 Helpful 0 Interesting 1 Brilliant 0 Confused

Is there any other possible value? Why did you say maximum possible value.

Siva Bathula - 3 years, 9 months ago

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No, there isn't any other value. I said maximum value to throw people off.

Sharky Kesa - 3 years, 9 months ago

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