# A Problem on Polynomials

Let $$x_1,x_2, x_3$$ be the roots of the equation $$x^3+3x+5 = 0$$. What is the value of the expression $$\displaystyle\Big(x_1+\frac{1}{x_1}\Big)$$$$\displaystyle\Big(x_2+\frac{1}{x_2}\Big)$$$$\displaystyle\Big(x_3+\frac{1}{x_3}\Big)$$?

One way in which we can do this is to break up the whole of $$\displaystyle\Big(x_1+\frac{1}{x_1}\Big)$$$$\displaystyle\Big(x_2+\frac{1}{x_2}\Big)$$$$\displaystyle\Big(x_3+\frac{1}{x_3}\Big)$$ and then just use the values we obtain from $$x^3+3x+5 = 0$$ by Vieta's formula. But, this is too long and it may get wrong somewhere. Does someone have a better method of doing this, maybe by transforming the equation??

Note by Dhrubajyoti Ghosh
10 months, 1 week ago

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The expression in question can be written as $$\dfrac{(x_1 ^2 + 1)(x_ 2 ^2 + 1)(x_ 3^2 + 1) }{x_1 x_2 x_3} = \dfrac{(x_1 ^2 + 1)(x_ 2 ^2 + 1)(x_ 3^2 + 1) }{-5}$$.

Hint: Let $$f(x) = x^3 + 3x + 5$$. What are the roots of $$f\left( \sqrt x \right) = 0$$?

Hint 2: If $$g(y)= 0$$ is a polynomial with roots $$y_1, y_2, y_3$$, then what roots does $$g(y - 1) = 0$$ has?

- 10 months, 1 week ago