Find the condition that the zeros of the polynomial f(x)=x3+3px2+3qx+r

1.	Find the condition that the zeros of the polynomial f(x)=x3+3px2+3qx+r may be in A.P.
Answer» The given quadratic polynomial is:f(x) = x3 + 3px2 + 3qx + rwe have to show that the zeroes of given polynomial are in the form of AP.Let, a - d, a, a + d be the zeroes of the polynomial, thenThe sum of zeroes = {tex}\\frac{{ - b}}{a}{/tex}a + a - d + a + d = -3p3a = - 3pa = - pSince, a is the zero of the polynomial f(x),Therefore, f(a) = 0f(a) = a3 + 3pa2 + 3qa + r = 0{tex}\\Rightarrow{/tex}\xa0a3 + 3pa2 + 3qa + r = 0{tex}\\Rightarrow{/tex}\xa0(-p)3 + 3p(-p)2 + 3q(-p) + r = 0{tex}\\Rightarrow{/tex}\xa0-p3 + 3p3 - 3pq + r = 0{tex}\\Rightarrow{/tex} 2p3 - 3pq + r = 0Which is the required condition.

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