Let `alpha`  and `beta`  be the roots of the equation `x^2+px+1=0`  and let `gamma`  and `delta`  be the roots of the equation `x^2+(1/p)x+1=0` . Show that A) `(alpha-gamma) (beta-gamma)...

Let `alpha`  and `beta`  be the roots of the equation `x^2+px+1=0`  and let `gamma`  and `delta`  be the roots of the equation `x^2+(1/p)x+1=0` .

Show that

A)

`(alpha-gamma) (beta-gamma) (alpha-delta) (beta-delta)= (gamma^2+pgamma+1) (delta^2+p delta+1)`

Asked on by roshan-rox

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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`alpha` and `beta` are roots of `x^2+px+1=0`

So we can say;

`(x-alpha) (x-beta) = x^2+px+1`

`(alpha-x) (beta-x) = x^2+px+1`

Let us say `x = gamma` ;

`(alpha-gamma) (beta-gamma) = gamma^2+p gamma+1------(1)`

Similarly if `x = delta` ; 

`(alpha-delta) (beta-delta) = delta^2+p delta+1----(2)`

`(1)xx(2)`

`(alpha-gamma) (beta-gamma) (alpha-delta) (beta-delta) = (gamma^2+p gamma+1) (delta^2+p delta+1)`

So the answer obtained as required.

`(alpha-gamma) (beta-gamma) (alpha-delta) (beta-delta) = (gamma^2+p gamma+1) (delta^2+p delta+1)`

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