Question

If `alpha,beta,gamma` and `delta` are roots of `x^4+9x^2+7x-8=0`, what is the value of `(alpha+beta)(gamma+delta)+alpha*beta+gamma*delta`?

Answer 1

`(alpha+beta)(gamma+delta)+alpha*beta+gamma*delta=9`

Explanation:

As `alpha,beta,gamma` and `delta` are roots of `x^4+9x^2+7x-8=0`

We have `(x-alpha)(x-beta)(x-gamma)(x-delta)hArrx^4+9x^2+7x-8=0`

or `x^4-(alpha+beta+gamma+delta)x^3+(alpha*beta+alpha*gamma+alpha*delta+beta*gamma+beta*delta+gamma*delta)x^2-(alpha*beta*gamma+alpha*gamma*delta+alpha*beta*delta+beta*gamma*delta)x+alphabetagammadeltahArrx^4+9x^2+7x-8` (A)

As `(alpha+beta)(gamma+delta)+alpha*beta+gamma*delta=alpha*gamma+alpha*delta+beta*gamma+beta*delta+alpha*beta+gamma*delta`

comparing the coefficients of `x^2` in (A)

`(alpha+beta)(gamma+delta)+alpha*beta+gamma*delta=9`