An algebra problem by Ezra Manatad
Algebra
Level
pending
Let X1, X2, x3 be the roots of the equation x^3+3x+5=0 . What is the value of the (x1+1/x1) (x2+1/x2)(x3+1/x3) ?
18/7
21/3
29/5
-29/5
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1 solution
By Vietas relation we already know that abc =-5, a+b+c=0 and ab+bc+ac=3
The required expression is equivalent to K= (a^2+1)(b^2+1)(c^2+1)/(abc). As mentioned above, the denominator of this expression is equal to -5
The numerator of K is simply equal to the product to the roots of a polynomial Q(x) with roots are a^2+1 , b^2+1, c^2+1.
Consider first a polynomial p(X) WHOSE ROOTS ARE A^2,B^2,C^2. doing some algebra we see,
a^2 b^2 +c^2=(a+b+c)=9
a^2 b^2 c^2=25
Thus, P9x)= x^3+6x^2+9x-25
The product of the roots of Q(x) ( which as mentioned is the numerator or K) is the negative of the constant term in Q(x), i.e. -Q(0).
-Q(0)=-(-1+6+-9-25)=29
K=29?9-5) = -29/5