First slide

Theory of expressions

Question

If roots of an equation $x^{n} - 1 = 0$ are $1, a_{1}, a_{2}, \dots, a_{n - 1}$ , then the value of $(1 - a_{1}) (1 - a_{2}) (1 - a_{3}) \dots (1 - a_{n - 1})$ will be

Moderate

A

n
B

n²
C

nⁿ
D

0

Solution

Clearly,
$\begin{array}{l} x^{n} - 1 = (x - 1) (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1}) \\ \Rightarrow \frac{x^{n} - 1}{x - 1} = (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1}) \\ \Rightarrow 1 + x + x^{2} + \dots + x^{n - 1} = (x - a_{1}) (x - a_{2}) \dots (x - a_{n - 1}) \\ \Rightarrow n = (1 - a_{1}) (1 - a_{2}) \dots (1 - a_{n - 1}) [putting x = 1] \end{array}$

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