p,a,r are the real roots of x^3-6x^2+3x+1=determine the possible values of p^q+q^r+r^2p

Answer 1

(x - p)(x - q)(x - r)

x³ - px² - qx² + prx + qrx + pqx - pqr

x³ - (p - q)x² + (pr + qr + pq)x - pqr

p - q = 6

pr + qr + pq = 3

pqr = -1

Now solve the system for p, q, and r and answer.

Answer 2

If p,q,r are the real roots of the cubic polynomial , then

(x-p) (x-q ) (x-r) = x³ -6 x² +3 x +1

Both side of this equation must agree about the coefficients of x³ , x² , x¹ and x⁰

For the left hand side, the coefficient of x can be worked out to be:

pq + pr + qr . On the right hand side the coefficient of x is 3, Thus

pq + pr + qr = 3.

I used a graphing calculator to find p , q and r , and indeed, the equation above checks out.