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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 $2a(a^2+4)(a+2)(a-2)$.
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