Find the condition that zeros pf …
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Sia ? 6 years, 4 months ago
Let a - d, a and a + d be the zeros of the polynomial F(x). Then,
Sum of the zeroes = {tex}- \frac { \text { Coefficient of } x ^ { 2 } } { \text { Coefficient of } x ^ { 3 } }{/tex}
{tex}\Rightarrow (a-d)+a+(a+d)=-\frac { ( - p ) } { 1 }{/tex}
{tex}\Rightarrow 3a=p{/tex}
{tex}\Rightarrow a=\frac { p } { 3 }{/tex}
Since {tex}a{/tex} is a zero of the polynomial {tex}f(x){/tex}. Therefore,
{tex}f(a)=0{/tex}
{tex}\Rightarrow a^3-pa^2+qa-r=0{/tex}
{tex}\Rightarrow \left( \frac { p } { 3 } \right) ^ { 3 } - p \left( \frac { p } { 3 } \right) ^ { 2 } + q \left( \frac { p } { 3 } \right) - r = 0{/tex}
{tex}\Rightarrow p^3-3p^3+9pq-27r=0{/tex}
{tex}\Rightarrow 2p^3-9pq+27r=0{/tex}
hence, {tex}2p^3-9pq+27r=0{/tex} is the required condition
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