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Question
Find the condition that the zeros of the polynomial f(x) = x3 + 3px2 + 3qx + r may be in A.P.
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Solution
Let a - d, a and a + d be the zeros of the polynomial f(x). Then,
Sum of the zeroes `=("coefficient of "x^2)/("coefficient of "x^3)`
`a-d+a+a+d=(-3p)/1`
`3a=-3p`
`a=(-3xxp)/3`
a = -p
Since 'a' is a zero of the polynomial f(x). Therefore,
f(x) = x3 + 3px2 + 3qx + r
f(a) = 0
f(a) = a3 + 3pa2 + 3qa + r
a3 + 3pa2 + 3qa + r = 0
Substituting a = -p we get,
(-p)3 + 3p(-p)2 + 3q(-p) + r = 0
-p3 + 3p3 - 3pq + r = 0
2p3 - 3pq + r = 0
Hence, the condition for the given polynomial is 2p3 - 3pq + r = 0
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