Question

factories :
x^4+x^2+1​

Answer 1

Answer:

The given eq. is x^4+x^2+1 ,

we know that , ( a+b )^2 = a^2 +b^2+ 2ab

to factorise the given eq. we need to calculate ( x^2+1)2 .

we should use formula (a+b) ^2= a^2+b^2+2ab to find value of (x^2+1) 2 .

Answer 2

$\large\underline{\sf{Solution-}}$

Given expression is

$\red{\rm :\longmapsto\: {x}^{4} + {x}^{2} + 1}$

To factorize this expression, add and subtract x², we get

$\rm \: = \: \: {x}^{4} + {x}^{2} + 1 + {x}^{2} - {x}^{2}$

$\rm \: = \: \: {x}^{4} + 2{x}^{2} + 1 - {x}^{2}$

$\rm \: = \: \: ({x}^{4} + 2{x}^{2} + 1) - {x}^{2}$

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

We know that,

$\blue{ \boxed{\bf \: {x}^{2} + 2xy + {y}^{2} = {(x + y)}^{2}}}$

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

Now, we further know that

$\blue{ \boxed{\bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

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

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

Hence,

Factorization of

$\red{\rm :\longmapsto\: {x}^{4} + {x}^{2} + 1}$

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

Let take one more example of same type!!

$\red{\rm :\longmapsto\: {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} }$

To factorize this expression, add and subtract x²y², we get

$\rm \: = \: \: {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} + {x}^{2} {y}^{2} - {x}^{2} {y}^{2}$

$\rm \: = \: \: {x}^{4} + 2{x}^{2} {y}^{2} + {y}^{4} - {x}^{2} {y}^{2}$

$\rm \: = \: \: ( {x}^{4} + 2{x}^{2} {y}^{2} + {y}^{4}) - {x}^{2} {y}^{2}$

$\rm \: = \: \: ( \: {( {x}^{2} )}^{2} + 2{x}^{2} {y}^{2} + {( {y}^{2} )}^{2} \: ) - {x}^{2} {y}^{2}$

We know,

$\blue{ \boxed{\bf \: {x}^{2} + 2xy + {y}^{2} = {(x + y)}^{2}}}$

$\rm \: = \: \: {( {x}^{2} + {y}^{2} ) }^{2} - {x}^{2} {y}^{2}$

$\rm \: = \: \: {( {x}^{2} + {y}^{2} ) }^{2} - ({x}{y})^{2}$

Now, we further know that,

$\blue{ \boxed{\bf \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

$\rm \: = \: \: ( {x}^{2}+ {y}^{2} + xy)( {x}^{2} + {y}^{2} - xy)$

Hence,

Factorization of

$\red{\rm :\longmapsto\: {x}^{4} + {x}^{2} {y}^{2} + {y}^{4} }$

$\red{\bf \: = \: \: ( {x}^{2}+ {y}^{2} + xy)( {x}^{2} + {y}^{2} - xy)}$

More Identities to know:

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

a² - b² = (a + b)(a - b)

(a + b)² = (a - b)² + 4ab

(a - b)² = (a + b)² - 4ab

(a + b)² + (a - b)² = 2(a² + b²)

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - b³ - 3ab(a - b)