# Factorize the following algebraic expressions:(i) $4x^4+1$(ii) $4x^4+y^4$

Given:

The given expressions are:

(i) $4x^4+1$.

(ii) $4x^4+y^4$

To do:

We have to factorize the given algebraic expressions.

Solution:

Factorizing algebraic expressions:

Factorizing an algebraic expression implies writing the expression as a product of two or more factors. Factorization is the reverse of distribution.

An algebraic expression is factored completely when it is written as a product of prime factors.

(i) The given expression is $4x^4+1$.

$4x^4+1$ can be written as,

$4x^4+1=4x^4+1+4x^2-4x^2$                    (Add and subtract $4x^2$)

$4x^4+1=[(2x^2)^2+2(2x^2)(1)+1^2]-4x^2$             [Since $4x^4=(2x^2)^2, 1=(1)^2$ and $4x^2=2(2x^2)(1)$]

Here, we can observe that the given expression is of the form $m^2+2mn+n^2$. So, by using the formula $(m+n)^2=m^2+2mn+n^2$, we can factorize the given expression.

Here,

$m=2x^2$ and $n=1$

Therefore,

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

$4x^4+1=(2x^2+1)^2-4x^2$

Now,

$(2x^2+1)^2-4x^2$ can be written as,

$(2x^2+1)^2-4x^2$=(2x^2+1)^2-(2x)^2$Using the formula$m^2-n^2=(m+n)(m-n)$, we can factorize$(2x^2+1)^2-(2x)^2$as,$4x^4+1=(2x^2+1)^2-(2x)^24x^4+1=(2x^2+1+2x)(2x^2+1-2x)$Hence, the given expression can be factorized as$(2x^2+2x+1)(2x^2-2x+1)$. (ii) The given expression is$4x^4+y^4$.$4x^4+y^4$can be written as,$4x^4+y^4=4x^4+y^4+4x^2y^2-4x^2y^2$(Add and subtract$4x^2y^2$)$4x^4+y^4=(2x^2)^2+(y^2)^2+2(2x^2)(y^2)-4x^2y^2$[Since$4x^4=(2x^2)^2, y^4=(y^2)^2$and$4x^2y^2=2(2x^2)(y^2)$] Here, we can observe that the given expression is of the form$m^2+2mn+n^2$. So, by using the formula$(m+n)^2=m^2+2mn+n^2$, we can factorize the given expression. Here,$m=2x^2$and$n=y^2$Therefore,$4x^4+y^4=(2x^2)^2+(y^2)^2+2(2x^2)(y^2)-4x^2y^24x^4+y^4=(2x^2+y^2)^2-4x^2y^2$Now,$(2x^2+y^2)^2-4x^2y^2$can be written as,$(2x^2+y^2)^2-4x^2y^2=(2x^2+y^2)^2-(2xy)^2$[Since$4x^2y^2=(2xy)^2$] Using the formula$m^2-n^2=(m+n)(m-n)$, we can factorize$(2x^2+y^2)^2-(2xy)^2$as,$(2x^2+y^2)^2-4x^2y^2=(2x^2+y^2)^2-(2xy)^2(2x^2+y^2)^2-4x^2y^2=(2x^2+y^2+2xy)(2x^2+y^2-2xy)$Hence, the given expression can be factorized as$(2x^2+y^2+2xy)(2x^2+y^2-2xy)\$.

Updated on: 11-Apr-2023

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