Let g(x)=x^4-5x^3+2x^2+7x-11, and let the roots of g(x) be p, q, r, and s.

Let   g(x)  =  x⁴ - 5x³ + 2x² + 7x - 11   with roots p, q, r, and s.

 

Since the roots are p, q, r, and s:   g(x)  =  (x - p)(x - q)(x - r)(x - s)

 

Multiply out  (x - p)(x - q)(x - r)(x - s)  

     =  x⁴ - x³p - x³​q - x³​r - x³​s + x²pq + x²​pr + x²​ps + x²​qr + x²​qs + x²​rs - xpqr - xpqs - xqrs - xprs - xqrs + pqrs

     =  x⁴ - (p + q + r + s)x³ + (pq + pr + ps + qr + qs + rs)x² - (pqr + pqs + qrs + prs)x + pqrs

 

Since the coefficients of the x-cubed terms are equal:  - (p + q + r + s)  =  - 5   --->   p + q + r + s  =  5

Since the coefficients of the x-squared terms are equal:  (pq + pr + ps + qr + qs + rs)  =  2

Since the coefficients of the x-terms are equal:  - (pqr + pqs + qrs + prs)  =  + 7   --->   pqr + pqs + qrs + prs  =  -7

and:  pqrs  =  -11

 

(a)  Compute pqr+pqs+prs+qrs  =  -7

 

(b)  Compute 1/p+1/q+1/r+1/s:  

       Write with the common denominator of pqrs:  (qrs + prs + pqs + pqr) / pqrs   =   -7/-11  =  7/11

 

(c)  Compute p^2+q^2+r^2+s^2:

          (p + q + r + s)²  =  p² + q² + r² + s² + 2pq + 2pr + 2ps + 2qr + 2qs + 2rs

                                   =  p² + q² + r² + s² + 2(pq + pr + ps + 2qr + 2qs + 2rs)

                                   =  p² + q² + r² + s² + 2(2)

                                   =  p² + q² + r² + s² + 4

          But, since  p + q + r + s  =  5

                --->     (5)²  =  p² + q² + r² + s² + 4

                --->      25  =  p² + q² + r² + s² + 4

                --->     p² + q² + r² + s²  =  21

 

(d)  Compute p^2qrs+pq^2rs+pqr^2s+pqrs^2:

          p^2qrs+pq^2rs+pqr^2s+pqrs^2  =  pqrs(p + q + r + s)  =  -11(p + q + r + s)  =  -11(5)  =  -55