Question

Show that x-4 is a factor of x¹0-1

Answer 1

x4−1 is a difference of squares

which factors in general as

∙xa2−b2=(a−b)(a+b)

here a=x2 and b=1

⇒x4−1=(x2−1)(x2+1)

x2−1 is a difference of squares

⇒x4−1=(x−1)(x+1)(x2+1)

we can factor x2+1 by equating to zero and solving

x2+1=0⇒x2=−1⇒x=±√−1=±i

factors are (x−(+i))(x−(−i))

⇒x4−1=(x−1)(x+1)(x−i)(x+i)