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Calculate the sum of the inverse roots of the equation $x^3-7x^2+4x-1=0$.

My development was: sum = $-7$ product = $1$ Thus, I believe that to find the inverse roots one only has to share the sum with the product, i.e. $-\frac{7}{1} = 7$, however, the answer given in the question is $4$. what do you think? do you have any formula?

Asaf Karagila
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funfun
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2 Answers2

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If we reverse the coefficients of a polynomial with roots $r_i$, we get a polynomial with roots $1/r_i$. In this case, the reversal is $-x^3+4x^2-7x+1=0$. By Viète's formulas, we see that the sum of reciprocals of roots of the original polynomial is $-\frac{4}{-1}=4$.

Parcly Taxel
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    For those who may not see why reversing the coefficients gives a polynomial with inverse roots, divide the original equation by its largest power of $x$. – Paul Sinclair Aug 08 '19 at 23:29
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Let the roots be $x_1,x_2,x_3$, all non-zero. Then you need$$\frac1{x_1}+\frac1{x_2}+\frac1{x_3}=\frac{x_1x_2+x_2x_3+x_3x_1}{x_1x_2x_3}=\frac{-c/a}{d/a}=-\frac cd$$where $x^3-7x^2+4x-1=ax^3+bx^2+cx+d$.

Shubham Johri
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