Question

Find the condition that the zeros of the polynomial f(x) = x³ + 3px² + 3qx + r may be in A.P.

Answer 1

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