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.)

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestPresumably, 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 \(g(-1)\).

Log in to reply

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 \(89 = (1+\alpha^2)\cdot(1+\beta^2)\cdot(1+\gamma^2)\cdot(1+\delta^2)\)

Log in to reply