# Factorize the expression $2a^5-32a$.

Given:

The given algebraic expression is $2a^5-32a$.

To do:

We have to factorize the expression $2a^5-32a$.

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.

$2a^5-32a$ can be written as,

$2a^5-32a=2a(a^4-16)$             (Taking $2a$ common)

$2a^5-32a=2a[(a^2)^2-4^2]$                  [Since $a^4=(a^2)^2, 16=4^2$]

Here, we can observe that the given expression is a difference of two squares. So, by using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize the given expression.

Therefore,

$2a^5-32a=2a[(a^2)^2-4^2]$

$2a^5-32a=2a(a^2+4)(a^2-4)$

Now,

$a^2-4$ can be written as,

$a^2-4=(a)^2-2^2$                         (Since $4=2^2$)

Using the formula $a^2-b^2=(a+b)(a-b)$, we can factorize $(a)^2-2^2$.

$(a)^2-2^2=(a+2)(a-2)$.............(I)

Therefore,

$2a^5-32a=2a(a^2+4)(a+2)(a-2)$

Hence, the given expression can be factorized as $2a(a^2+4)(a+2)(a-2)$.

Updated on: 09-Apr-2023

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