Question

Factorise: x4 +x2 +1

Answer 1

Factorise this expression using the difference of squares: 
f(x) = x⁴ + x² + 1 
f(x) = (x² + 1)² - x² 
f(x) = (x² + 1 - x)(x² + 1 + x) 
f(x) = (x² - x + 1)(x² + x + 1) 
f(x) = (x² + x + 1)(x² - x + 1)

Answer 2

x⁴ + x² + 1 = (x² + x + 1)(x² - x + 1)

Given :

The expression x⁴ + x² + 1

To find :

To factorise the expression

Formula :

• (a + b)² = a² + 2ab + b²

• a² - b² = ( a + b ) ( a - b )

Solution :

Step 1 of 2 :

Write down the given expression

The given expression is

x⁴ + x² + 1

Step 2 of 2 :

Factorise the expression

We are aware of the identity that

a² - b² = ( a + b ) ( a - b )

Thus we get ,

$\displaystyle \sf{ {x}^{4} + {x}^{2} + 1}$

$\displaystyle \sf{ = {x}^{4} + 2 {x}^{2} - {x}^{2} + 1}$

$\displaystyle \sf{ = {x}^{4} + 2 {x}^{2} + 1 - {x}^{2} }$

$\displaystyle \sf{ = {( {x}^{2} )}^{2} + 2. {x}^{2} .1+ {(1)}^{2} - {x}^{2} }$

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

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

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

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

∴ x⁴ + x² + 1 = (x² + x + 1)(x² - x + 1)

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