Find the condition that the zeros …
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Sia ? 6 years, 6 months ago
The given quadratic polynomial is:
f(x) = x3 + 3px2 + 3qx + r
we 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, then
The sum of zeroes = {tex}\frac{{ - b}}{a}{/tex}
a + a - d + a + d = -3p
3a = - 3p
a = - p
Since, a is the zero of the polynomial f(x),
Therefore, f(a) = 0
f(a) = a3 + 3pa2 + 3qa + r = 0
{tex}\Rightarrow{/tex} a3 + 3pa2 + 3qa + r = 0
{tex}\Rightarrow{/tex} (-p)3 + 3p(-p)2 + 3q(-p) + r = 0
{tex}\Rightarrow{/tex} -p3 + 3p3 - 3pq + r = 0
{tex}\Rightarrow{/tex} 2p3 - 3pq + r = 0
Which is the required condition.
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