If α is any of $$7th$$ roots of unity, then α$$=$$ cis $$2K$$$$π$$$$7(K=0$$ to $$6)$$ and ∑$$i=16$$ α$$i=$$−$$1($$α≠$$1)$$ and α$$7=1$$. Then answer the following The equation whose roots are α$$+$$α$$2+$$α$$4$$ and α$$3+$$α$$5+$$α$$6$$ is $$($$α≠$$1)$$ If $$f(x)=1+2x+3x2+4x3+5x4+6x5+7x6$$ then for α≠$$1$$, $$f(x)+f($$α$$x)+f$$α$$2x+f$$α$$3x+$$−−−$$+f$$α$$6x=$$

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If α is any of $$7th$$  roots of unity, then α$$=$$ cis $$2K$$$$π$$$$7(K=0$$ to $$6)$$ and ∑$$i=16$$ α$$i=$$−$$1($$α≠$$1)$$ and α$$7=1$$. Then answer the following  The equation whose roots are α$$+$$α$$2+$$α$$4$$ and α$$3+$$α$$5+$$α$$6$$ is $$($$α≠$$1)$$ If $$f(x)=1+2x+3x2+4x3+5x4+6x5+7x6$$ then for α≠$$1$$, $$f(x)+f($$α$$x)+f$$α$$2x+f$$α$$3x+$$−−−$$+f$$α$$6x=$$

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Let α$$=cis$$⁡$$2$$$$π$$$$7$$ then $$a=$$α$$+$$α$$2+$$α$$4$$,$$b=$$α$$3+$$α$$5+$$α$$6$$⇒$$a+b=$$α$$+$$α$$2+$$α$$3+$$α$$4+$$α$$5+$$α$$6=$$−$$1$$ & $$ab=$$α$$+$$α$$2+$$α$$4$$α$$3+$$α$$5+$$α$$6=2$$∴ Equation whose roots are a,b is $$x2$$−$$x($$−$$1)+2=0$$ $$f(x)+f($$α$$x)+f$$α$$2x+f$$α$$3x+$$−−−$$+f$$α$$6x=(1+1+1$$−−$$+1)+2x1+$$α$$+$$α$$2+$$α$$3+$$−−−$$+$$α$$6+3x21+$$α$$2+$$α$$4+$$−−−$$+$$α$$12+$$−−−−$$+=7x61+$$α$$6+$$α$$12+$$α$$18+$$−−−$$+$$α$$36=7$$

If α is any of 7th roots of unity, then α= cis 2Kπ7(K=0 to 6) and ∑i=16 αi=−1(α≠1) and α7=1. Then answer the following The equation whose roots are α+α2+α4 and α3+α5+α6 is (α≠1) If f(x)=1+2x+3x2+4x3+5x4+6x5+7x6 then for α≠1, f(x)+f(αx)+fα2x+fα3x+−−−+fα6x=

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