Find all roots of the polynomial equation&nbsp;p(p(x))−x=0.Let p(x) be a quadratic polynomial such that for distinct reals&nbsp;α&nbsp;and&nbsp;β,p(α)=α&nbsp;&amp;&nbsp;p(β)=βShow that&nbsp;α&nbsp;and&nbsp;β&nbsp;are the roots of the following equationp(p(x))−x=0

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Find all roots of the polynomial equation&amp;nbsp;p(p(x))−x=0.Let p(x) be a quadratic polynomial such that for distinct reals&amp;nbsp;α&amp;nbsp;and&amp;nbsp;β,p(α)=α&amp;nbsp;&amp;amp;&amp;nbsp;p(β)=βShow that&amp;nbsp;α&amp;nbsp;and&amp;nbsp;β&amp;nbsp;are the roots of the following equationp(p(x))−x=0

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Write&amp;nbsp;q(x)=p(x)−x, then given equation is equivalent toq(q(x)+x)+q(x)=0Since&amp;nbsp;α&amp;nbsp;and&amp;nbsp;β&amp;nbsp;are roots for q we haveq(x)=c(x−α)(x−β)where&amp;nbsp;c≠0, soc(q(x)+x−α)(q(x)+x−β)+c(x−α)(x−β)=0so&amp;nbsp;(x−α)(x−β)((cx−cα+1)(cx−cβ+1)+1)=0Need to solve the red equation...

Find all roots of the polynomial equation

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