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Polynomial Problem in Pre-RMO today !

Given a polynomial :

\[ p(x) = x^5 - 3x^4 + 5x^3 - 2x^2 + 9x - 7 = 0 \]

With \( \alpha , \beta , \gamma \) and \( \sigma \) as its roots .

Find :

\[ (1 + \alpha^2)(1+\beta^2)(1+\gamma^2)(1+\sigma^2) \]

Please help .

(Not pretty sure about some coefficients, but the idea is same.)

Note by Priyansh Sangule
4 years ago

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Presumably, you mean the quintic polynomial to have roots \(\alpha,\beta,\gamma,\delta,\epsilon\), and want to know \[ (1+\alpha^2)(1+\beta^2)(1+\gamma^2)(1+\delta^2)(1+\epsilon^2) \] Find the monic quintic polynomial \(g(y)\) whose roots are \(\alpha^2,\beta^2,\gamma^2,\delta^2,\epsilon^2\). Do this by substituting \(x=\sqrt{y}\) and eliminating the square root. Then consider \(g(-1)\).

Mark Hennings - 4 years ago

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Let \(p(x) = x^5-3x^4+5x^3-2x^2+9x-7 = (x-\alpha)\cdot(x-\beta)\cdot(x-\gamma)\cdot (x-\delta)\)

Now put \(x = i\) and \(x= -i\) respectively

\(i^5-3i^4+5i^3-2i^2+9i-7 = (5i-8) = (\alpha - i)\cdot(\beta-i)\cdot(\gamma-i)\cdot(\delta-i)\)

\(-i^5-3i^4-5i^3-2i^2-9i-7 = -(5i-8) = (\alpha + i)\cdot(\beta+i)\cdot(\gamma+i)\cdot(\delta+i)\)

Now multiply these two, we get \(89 = (1+\alpha^2)\cdot(1+\beta^2)\cdot(1+\gamma^2)\cdot(1+\delta^2)\)

Jagdish Singh - 4 years ago

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