Question

factorise
${x}^{5} - 1$
​

Answer 1

${\huge {\mathfrak {Answer :-}}}$

✨${} x^5 - 1$

${} = x^5 - x^2 + x^2 - 1$

${} = x^2(x^3 - 1) + (x + 1)(x - 1)$

${} = x^2(x - 1)(x^2 + x + 1) + (x + 1)(x - 1)$

${} = (x - 1)\: [ \: x^2(x^2 + x + 1) + x + 1) \:$

${} = (x - 1)(x^4 + x^3 +x^2 + x + 1)$

Done.. ✔️

Thanks..

Answer 2

Answer:

Step-by-step explanation:

Given :

To factorize $x^5 - 1$

Solution :

We know that,

Identities (used in this problem),

$(x^2 - y^2 = (x + y)( x - y)$   ..(i)

$x^3 - y^3 = (x - y)(x^2 - xy + y^2)$  ..(ii)

_

$x^5 - 1$

⇒ $x^5 - x^2 + x^2 - 1$

⇒ $x^2(x^3 - 1) + (x^2 - 1)$

⇒ $x^2(x^3 - 1) + (x - 1)(x + 1)$ by (i)

⇒ $x^2(x - 1)(x^2 + x + 1) + (x - 1)(x + 1)$ by (ii)

⇒ $(x - 1)[x^2(x^2 + x + 1) + (x + 1)]$

⇒ $(x - 1)[(x^4 + x^3 + x^2) + (x + 1)]$

⇒ $(x - 1)(x^4 + x^3 + x^2 + x + 1)$

∴  $x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1)$