If zeroes of the polynomials #f(x)=x^3-3px^2+qx-r# are in AP then what is the relation between p, q and r?
1 Answer
Jan 13, 2018
Explanation:
Denoting the three zeros by
#f(x) = x^3-3px^2+qx-r#
#color(white)(f(x)) = (x-alpha)(x-beta)(x-gamma)#
#color(white)(f(x)) = x^3-(alpha+beta+gamma)x^2+(alphabeta+betagamma+gammaalpha)x-alphabetagamma#
In particular, equating the coefficient of
#alpha+beta+gamma = 3p#
If
#alpha = beta-delta#
#gamma = beta+delta#
So:
#3 beta = (beta-delta)+beta+(beta+delta) = alpha+beta+gamma = 3p#
So:
#beta = p#
That is:
So:
#0 = f(p) = color(blue)(p^3)-3pcolor(blue)(p)^2-qcolor(blue)(p)-r = -2p^3-pq-r#
Inverting the signs, that is:
#2p^3+pq+r = 0#