Question

1S
Show that (x - 1) is a
factor of x power n -1
-​

Answer 1

$\bigstar$ Question:

• Show that ( x - 1 ) is a factor of ( x^n - 1 )

$\bigstar$ To show that:

• $(x - 1) \: is \: a \: factor \: of \: \\ ( {x}^{n} - 1)$

$\bigstar$ Solution:

• Here, we will use mid - point theorem to proof this statement.

Firstly, ( x - 1 ) = 0

$\implies$ x = (1)

Now, p(x)

$= ({x}^{n} - 1)$

$\implies$ p(1)

$= ( {1}^{n} - 1)$

( We know that any power of 1 will be always 1 , it does matter if the power is negative or positive because (1)^ (-1) = 1 and 1¹ = 1)

$= 1 - 1$

$= 0$

Since we get zero, then ( x - 1) is a factor of

(x^n - 1).

$\bigstar$ Answer:

• Therefore, it is proved that x - 1 is a factor of this polynomial.

Answer 2

To FinD :-

• $(x - 1) \: is \: a \: fector \: of \: ( {x}^{n} - 1)$

Solution :-

$x - 1 = 0 \\ x = 1$

Now, let's use the mid term theorem

$p(x) = {x}^{n} - 1$

$p(1) = ( {1)}^{n} - 1$

We know that the power of 1 is always 1

$= 1 - 1 \\ \\ = 0$

Hence proved that, (x - 1) is a factor of xⁿ - 1