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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,
$4x^4+1=(2x^2+1)^2-(2x)^2$
$4x^4+1=(2x^2+1+2x)(2x^2+1-2x)$
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,
$4x^4+y^4=(2x^2)^2+(y^2)^2+2(2x^2)(y^2)-4x^2y^2$
$4x^4+y^4=(2x^2+y^2)^2-4x^2y^2$
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 $(2x^2+y^2+2xy)(2x^2+y^2-2xy)$.
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