# 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)`

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### 1 Answer

`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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