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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
5 months 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?

Pi Han Goh - 5 months ago

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