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