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if 1 a a1 a2 a3.........................a(n-1) are the n th roots of unity then (1-a1)x(1-a2)X(1-a3)...............................(1-a(n-1))=?

if 1 a a1 a2 a3.........................a(n-1) are the n th roots of unity then (1-a1)x(1-a2)X(1-a3)...............................(1-a(n-1))=?

Grade:12

1 Answers

jagdish singh singh
173 Points
7 years ago
$Given $\bf{x=1,a_{1},a_{2},a_{3},..........,a_{n-1}}$ are roots of $\bf{x=(1)^{\frac{1}{n}}\Rightarrow x^n-1=0}$\\\\ So we get using factor Theorem $\bf{(x-1),(x-a_{1}),(x-a_{2}),........(x-a_{n-1})}$\\\\ are factor of $\bf{x^n-1}$\\\\ So $\bf{x^n-1 = (x-1)\cdot (x-a_{1})\cdot (x-a_{2}),........(x-a_{n-1})}$\\\\So $\bf{\frac{x^n-1}{x-1}= (x-a_{1})\cdot (x-a_{2}),........(x-a_{n-1})}$
 
\hspace{-0.5 cm}$So $\bf{\lim_{x\rightarrow 1}\frac{x^n-1}{x-1} = \lim_{x\rightarrow 1}(x-a_{1})\cdot (x-a_{2})\cdot ......(x-a_{n-1})}$\\\\ Now Apply H, Hopital Rule, \\\\ So we get $\bf{n=(1-a_{1})\cdot (1-a_{2})\cdot ......(1-a_{n-1}).}$

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